Further results on the least Q-eigenvalue of a graph with fixed domination number

TL;DR

This paper investigates the least Q-eigenvalue of connected nonbipartite graphs with fixed order and domination number, providing exact results for specific domination numbers and odd-girth constraints.

Contribution

It determines the minimum Q-eigenvalue for nonbipartite graphs with given domination number and odd-girth, extending previous spectral graph theory results.

Findings

Minimum Q-eigenvalue for graphs with domination number (n-1)/2 is fully characterized.

Minimum Q-eigenvalue for graphs with odd-girth ≤ 5 and certain domination numbers is fully characterized.

Provides new spectral bounds related to domination number and odd-girth in nonbipartite graphs.

Abstract

In this paper, we proceed on determining the minimum $q_{min}$ among the connected nonbipartite graphs on $n\geq 5$ vertices and with domination number $\frac{n+1}{3}<\gamma\leq \frac{n-1}{2}$. Further results obtained are as follows: $\mathrm{(i)}$ among all nonbipartite connected graph of order $n\geq 5$ and with domination number $\frac{n-1}{2}$, the minimum $q_{min}$ is completely determined; $\mathrm{(ii)}$ among all nonbipartite graphs of order $n\geq 5$, with odd-girth $g_{o}\leq5$ and domination number at least $\frac{n+1}{3}<\gamma\leq \frac{n-2}{2}$, the minimum $q_{min}$ is completely determined.

Equations

$$\sum _{i=1}^k{\tau }_{i_j,i_{j+1}}=\sum _{{\tau }_{i_s,i_{s+1}}\ne 0}{\tau }_{i_s,i_{s+1}}\le \lceil \frac{{\sum }_{{\tau }_{i_s,i_{s+1}}\ne 0}(i_{s+1}-i_s-3)}{3}\rceil \le \lceil \frac{g-k-2}{3}\rceil .$$

Full text

$\mathrm{(i)}$ among all nonbipartite connected graph of order $n\geq 5$ and with domination number $\frac{n-1}{2}$, the minimum $q_{min}$ is completely determined;

$\mathrm{(ii)}$ among all nonbipartite graphs of order $n\geq 5$, with odd-girth $g_{o}\leq 5$ and domination number at least $\frac{n+1}{3}<\gamma\leq\frac{n-2}{2}$, the minimum $q_{min}$ is completely determined.

AMS Classification: 05C50

Keywords: Domination number; Signless Laplacian; Nonbipartite graph; Least eigenvalue

1 Introduction

All graphs considered in this paper are connected, undirected and simple, i.e., no loops or multiple edges are allowed. We denote by $\parallel S\parallel$ the $cardinality$ of a set $S$, and denote by $G=G[V(G)$, $E(G)]$ a graph with vertex set $V(G)=\{v_{1},v_{2},\ldots,v_{n}\}$ and edge set $E(G)$ where $\parallel V(G)\parallel=n$ is the $order$ and $\parallel E(G)\parallel=m$ is the $size$.

In a graph, if vertices $v_{i}$ and $v_{j}$ are adjacent (denoted by $v_{i}\sim v_{j}$), we say that they $dominate$ each other. A vertex set $D$ of a graph $G$ is said to be a $dominating$ $set$ if every vertex of $V(G)\setminus D$ is adjacent to (dominated by) at least one vertex in $D$. The $domination$ $number$ $\gamma(G)$ ($\gamma$, for short) is the minimum cardinality of all dominating sets of $G$. For a graph $G$, a dominating set is called a $minimal$ $dominating$ $set$ if its cardinality is $\gamma(G)$. A well known result about $\gamma(G)$ is that for a graph $G$ of order $n$ containing no isolated vertex, $\gamma\leq\frac{n}{2}$ [12]. A comprehensive study of issues relevant to dominating set of a graph has been undertaken because of its good applications [8], [19].

Recall that $Q(G)=D(G)+A(G)$ is called the $signless$ $Laplacian$ $matrix$ (or $Q$-$matrix$) of $G$, where $D(G)=\mathrm{diag}(d_{1},d_{2},\ldots,d_{n})$ with $d_{i}=d_{eg}(v_{i})$ being the degree of vertex $v_{i}$ $(1\leq i\leq n)$, and $A(G)$ is the adjacency matrix of $G$. The signless Laplacian has attracted the attention of many researchers and it is being promoted by many researchers [1], [2]-[6], [15].

The least eigenvalue of $Q(G)$, denote by $q_{min}(G)$ or $q_{min}$, is called the $least$ $Q$-$eigenvalue$ of $G$. Because $Q(G)$ is positive semi-definite, we have $q_{min}(G)\geq 0.$ From [2], we know that, for a connected graph $G$, $q_{min}(G)=0$ if and only if $G$ is bipartite. Consequently, in [7], $q_{min}$ was studied as a measure of nonbipartiteness of a graph. One can notice that there are quite a few results about $q_{min}$. In [1], D.M. Cardoso et al. determined the graphs with the the minimum $q_{min}$ among all the connected nonbipartite graphs with a prescribed number of vertices. In [6], L. de Lima et al. surveyed some known results about $q_{min}$ and also presented some new results. In [9], S. Fallat, Y. Fan investigated the relations between $q_{min}$ and some parameters reflecting the graph bipartiteness. In [15], Y. Wang, Y. Fan investigated $q_{min}$ of a graph under some perturbations, and minimized $q_{min}$ among the connected graphs with fixed order which contains a given nonbipartite graph as an induced subgraph. Recently, in [14], the authors determined all non-bipartite hamiltonian graphs whose $q_{min}$ attains the minimum.

Recall that a $lollipop$ $graph$ $L_{g,l}$ is a graph composed of a cycle $\mathbb{C}=v_{1}v_{2}\cdots v_{g}v_{1}$ and a path $\mathbb{P}=v_{g}v_{g+1}\cdots v_{g+l}$ with $l\geq 1$. For given $g$ and $l$, a graph of order $n$ is called a $F_{g,l}$-$graph$ if it is obtained by attaching $n-g-l$ pendant vertices to some nonpendant vertices of a $L_{g,l}$. If $l=1$, a $F_{g,l}$-graph is also called a $sunlike$ graph. In a graph, a vertex is called a $p$-$dominator$ (or $support$ $vertex$) if it dominates a pendant vertex. In a $F_{g,l}$-graph if each $p$-dominator other than $v_{g+l-1}$ is attached with exactly one pendant vertex, then this graph is called a $\mathcal{F}_{g,l}$-$graph$. A $\mathcal{F}_{g,l}$-graph is called a $\mathcal{F}^{\circ}_{g,l}$-$graph$ if $v_{g}$ is a $p$-dominator. In the following paper, for unity, for a $\mathcal{F}_{g,l}$-graph, $\mathbb{C}$ and $\mathbb{P}$ are expressed as above.

v_{2}$$v_{3}$$v_{4}$$v_{1}$$v_{a_{1}}$$v_{a_{2}}$$v_{a_{k}}$$v_{\varepsilon-k-2}$$v_{\varepsilon-k-3}$$v_{2}$$v_{3}$$v_{1}Fig. 1.1. $\mathcal{H}^{k}_{1}$, \mathcal{H}^{k}_{2}$$\mathcal{H}^{k}_{1}$$\mathcal{H}^{k}_{2}$$v_{\varepsilon}$$v_{\varepsilon-1}$$v_{\varepsilon-1}$$v_{\varepsilon}$$v_{\varepsilon-k-1}$$v_{\varepsilon-2}

Let $\mathcal{H}^{k}_{1}$ be a $\mathcal{F}_{3,\varepsilon-3}$-graph of order $n\geq 4$ where there are $k\geq 0$ $p$-dominators among $v_{1}$, $v_{2}$, $\ldots$, $\varepsilon-2$ ($\varepsilon\geq 3$. see Fig. 1.1). If $k\geq 1$, in $\mathcal{H}^{k}_{1}$, suppose $v_{a_{j}}s$ are $p$-dominators where $1\leq j\leq k$, $1\leq a_{1}<a_{2}<\cdots<a_{k}\leq\varepsilon-2$, and suppose $v_{\tau_{j}}$ is the pendant vertex attached to $v_{a_{j}}$. Let $\mathcal{H}^{k}_{2}=\mathcal{H}^{k}_{1}-\sum\limits_{j=1}^{k}v_{\tau_{j}}v_{a_{j}}+\sum\limits_{j=1}^{k}v_{\tau_{j}}v_{\varepsilon-2-k+j}$ (see Fig. 1.1). If $k=0$, then $\mathcal{H}^{0}_{1}=\mathcal{H}^{0}_{2}$. If $\alpha\geq 1$, we denoted by $\mathscr{H}_{3,\alpha}$ the graph $\mathcal{H}^{\alpha-1}_{2}$ of order $n$ in which there are $\alpha$ $p$-dominators and $v_{\varepsilon-1}$ has only one pendant vertex (where $\varepsilon=n-\alpha+1$); if $\alpha=0$, we let $\mathscr{H}_{3,0}=C_{3}=v_{1}v_{2}v_{3}v_{1}$.

In [10] and [17], the authors first considered the relation between $q_{min}$ of a graph and its domination number. Among all the nonbipartite graphs with both order $n\geq 4$ and domination number $\gamma\leq\frac{n+1}{3}$, they characterized the graphs with the minimum $q_{min}$. A remaining open problem is that how about the $q_{min}$ of the connected nonbipartite graph on $n$ vertices with domination number $\frac{n+1}{3}<\gamma\leq\frac{n}{2}$. In [18], the authors proceeded on considering this problem. Among the nonbipartite graphs of order $n=4$, the minimum $q_{min}$ is completely determined; among the nonbipartite graphs of order $n$ and with given domination number $\frac{n}{2}$, the minimum $q_{min}$ is completely determined; further results about the domination number, the $q_{min}$ of a graph as well as their relation are represented. An open problem still left is that how to determine the minimum $q_{min}$ of the connected nonbipartite graph on $n\geq 5$ vertices with domination number $\frac{n+1}{3}<\gamma\leq\frac{n-1}{2}$. Let $\mathbb{S}=\mathscr{H}_{3,\alpha}$ be of order $n\geq 4$ where $\alpha$ is the least integer such that $\lceil\frac{n-2\alpha-2}{3}\rceil+\alpha=\gamma$. In [18], the authors represented some structural characterizations about the minimum $q_{min}$ for this problem, and conjectured that such $\mathbb{S}$ has the smallest $q_{min}$. However, the problem seems really difficult to solve. Motivated by proceeding on solving this problem, we go on with our research and get some further results as follows.

Theorem 1.1

Let $G$ be a nonbipartite connected graph of order $n\geq 5$ and with domination number $\frac{n-1}{2}$. Then $q_{min}(G)\geq q_{min}(\mathscr{H}_{3,\frac{n-1}{2}})$ with equality if and only if $G\cong\mathscr{H}_{3,\frac{n-1}{2}}$.

Theorem 1.2

Among all nonbipartite graphs of order $n\geq 5$, with odd-girth $g_{o}\leq 5$ (length of the shortest odd cycle in this graph) and domination number $\frac{n+1}{3}<\gamma\leq\frac{n-2}{2}$, then the least $q_{min}$ attains the minimum uniquely at a $\mathscr{H}_{3,\alpha}$ where $\alpha\leq\frac{n-3}{2}$ is the least integer such that $\lceil\frac{n-2\alpha-2}{3}\rceil+\alpha=\gamma$.

2 Preliminary

In this section, we introduce some notations and some working lemmas.

Denote by $P_{n}$, $C_{n}$, $K_{n}$, a $path$, a $n$-cycle (of length $n$), a $complete$ graph of order $n$ respectively. If $k$ is odd, we say $C_{k}$ an $odd$ $cycle$. The $girth$ of a graph $G$, denoted by $g$, is the length of the shortest cycle in $G$. The $odd$-$girth$ for a nonbipartite graph $G$, denoted by $g_{o}(G)$ or $g_{o}$, is the length of the shortest odd cycle in this graph. $G-v_{i}v_{j}$ denotes the graph obtained from $G$ by deleting the edge $v_{i}v_{j}\in E(G)$, and let $G-v_{i}$ denote the graph obtained from $G$ by deleting the vertex $v_{i}$ and the edges incident with $v_{i}$. Similarly, $G+v_{i}v_{j}$ is the graph obtained from $G$ by adding an edge $v_{i}v_{j}$ between its two nonadjacent vertices $v_{i}$ and $v_{j}$. Given an vertex set $S$, $G-S$ denotes the graph obtained by deleting all the vertices in $S$ from $G$ and the edges incident with any vertex in $S$.

A connected graph $G$ of order $n$ is called a $unicyclic$ graph if $\|E(G)\|=n$. For $S\subseteq V(G)$, let $G[S]$ denote the subgraph induced by $S$. Denoted by $d_{istG}(v_{i},v_{j})$ the $distance$ between two vertices $v_{i}$ and $v_{j}$ in a graph $G$.

For a graph $G$ of order $n$, let $X=(x_{1},x_{2},\ldots,x_{n})^{T}\in R^{n}$ be defined on $V(G)$, i.e., each vertex $v_{i}$ is mapped to the entry $x_{i}$; let $|x_{i}|$ denote the $absolute$ $value$ of $x_{i}$. One can find that $X^{T}Q(G)X=\sum_{v_{i}v_{j}\in E(G)}(x_{i}+x_{j})^{2}.$ In addition, for an arbitrary unit vector $X\in R^{n}$, $q_{min}(G)\leq X^{T}Q(G)X$, with equality if and only if $X$ is an eigenvector corresponding to $q_{min}(G)$.

Lemma 2.1

[3]* Let $G$ be a graph on $n$ vertices and $m$ edges, and let $e$ be an edge of $G$. Let $q_{1}\geq q_{2}\geq\cdots\geq q_{n}$ and $s_{1}\geq s_{2}\geq\cdots\geq s_{n}$ be the $Q$-eigenvalues of $G$ and $G-e$ respectively. Then $0\leq s_{n}\leq q_{n}\leq\cdots\leq s_{2}\leq q_{2}\leq s_{1}\leq q_{1}.$*

Let $G_{1}$ and $G_{2}$ be two disjoint graphs, and let $v_{1}\in V(G_{1}),$ $v_{2}\in V(G_{2})$. The $coalescence$ of $G_{1}$ and $G_{2}$, denoted by $G_{1}(v_{1})\diamond G_{2}(v_{2})$ or $G_{1}(u)\diamond G_{2}(u)$, is obtained from $G_{1}$, $G_{2}$ by identifying $v_{1}$ with $v_{2}$ and forming a new vertex $u$ where for $i=1,2$, $G_{i}$ can be trivial (that is, $G_{i}$ is only one vertex). For a connected graph $G=G_{1}(u)\diamond G_{2}(u)$, $i=1$, $2$, $G_{i}$ is called a $branch$ of $G$ with root $u$. For a vector $X=(x_{1},x_{2},\ldots,x_{n})^{T}\in R^{n}$ defined on $V(G)$, a branch $H$ of $G$ is called a $zero$ $branch$ with respect to $X$ if $x_{i}=0$ for all $v_{i}\in V(H)$; otherwise, it is called a $nonzero$ $branch$ with respect to $X$.

Lemma 2.2

[15]* Let $G$ be a connected graph which contains a bipartite branch $H$ with root $v_{s}$, and let $X$ be an eigenvector of $G$ corresponding to $q_{min}(G)$.*

(i)* If $x_{s}=0$, then $H$ is a zero branch of G with respect to $X$;*

(ii)* If $x_{s}\neq 0$, then $x_{p}\neq 0$ for every vertex $v_{p}\in V(H)$. Furthermore, for every vertex $v_{p}\in V(H)$, $x_{p}x_{s}$ is either positive or negative depending on whether $v_{p}$ is or is not in the same part of the bipartite graph $H$ as $v_{s}$; consequently, $x_{p}x_{t}<0$ for each edge $v_{p}v_{t}\in E(H)$.*

Lemma 2.3

[15]* Let $G$ be a connected nonbipartite graph of order $n$, and let $X$ be an eigenvector of $G$ corresponding to $q_{min}(G)$. $T$ is a tree which is a nonzero branch of $G$ with respect to $X$ and with root $v_{s}$. Then $|x_{t}|<|x_{p}|$ whenever $v_{p},$ $v_{t}$ are vertices of $T$ such that $v_{t}$ lies on the unique path from $v_{s}$ to $v_{p}$.*

Lemma 2.4

[16]* Let $G=G_{1}(v_{2})\diamond T(u)$ and $G^{*}=G_{1}(v_{1})\diamond T(u)$, where $G_{1}$ is a connected nonbipartite graph containing two distinct vertices $v_{1},v_{2}$, and $T$ is a nontrivial tree. If there exists an eigenvector $X=(\,x_{1}$, $x_{2}$, $\ldots$, $x_{k}$, $\ldots)^{T}$ of $G$ corresponding to $q_{min}(G)$ such that $|x_{1}|>|x_{2}|$ or $|x_{1}|=|x_{2}|>0$, then $q_{min}(G^{*})<q_{min}(G)$.*

Lemma 2.5

[16]* Let $G=C(v_{0})\diamond B(v_{0})$ be a graph of order $n$, where C=v_{0}$$v_{1}$$v_{2} $\cdots$ $v_{2k}$ is a cycle of length $2k+1$, and $B$ is a bipartite graph of order $n-2k$. Then there exists an eigenvector $X=(\,x_{0}$, $x_{1}$, $x_{2}$, $\ldots$, $x_{2k}$ $\,)^{T}$ corresponding to $q_{min}(G)$ satisfying the following:*

(i)* $|x_{0}|=\max\{|x_{i}|\,|\,v_{i}\in V(C)\}>0$;*

(ii)* $x_{i}=x_{2k-i+1}$ for $i=1,2,\ldots,k$;*

(iii)* $x_{i}x_{i-1}\leq 0$ for $i=1,2,\ldots,k$, $x_{2k}x_{0}\leq 0$ and $x_{2k-i+1}x_{2k-i+2}\leq 0$ for $i=2,\ldots,k$.*

Moreover, if $2k+1<n$, then the multiplicity of $q_{min}(G)$ is one, and then any eigenvector corresponding to $q_{min}(G)$ satisfies (i), (ii), (iii).

Lemma 2.6

[5]* Let $G$ be a connected graph of order $n$. Then $q_{min}<\delta$, where $\delta$ is the minimal vertex degree of $G$.*

Lemma 2.7

[17]* Let $G$ be a nonbipartite graph with domination number $\gamma(G)$. Then $G$ contains a nonbipartite unicyclic spanning subgraph $H$ with both $g_{o}(H)=g_{o}(G)$ and $\gamma(H)=\gamma(G)$.*

Lemma 2.8

[17]* Suppose a graph $G$ contains pendant vertices. Then*

$\mathrm{(i)}$* there must be a minimal dominating set of $G$ containing all of its $p$-dominators but no any pendant vertex;*

$\mathrm{(ii)}$* if $v$ is a $p$-dominator of $G$ and at least two pendant vertices are adjacent to $v$, then any minimal dominating set of $G$ contains $v$ but no any pendant vertex adjacent to $v$.*

Lemma 2.9

[11]* (i) For a path $P_{n}$, we have $\gamma(P_{n})=\lceil\frac{n}{3}\rceil$.*

(ii) For a cycle $C_{n}$, we have $\gamma(C_{n})=\lceil\frac{n}{3}\rceil$.

We define the corona $G$ of graphs $G_{1}$ and $G_{2}$ as follows. The $corona$ $G=G_{1}\circ G_{2}$ is the graph formed from one copy of $G_{1}$ and $\parallel V(G_{1})\parallel$ copies of $G_{2}$ where the $i$th vertex of $G_{1}$ is adjacent to every vertex in the $i$th copy of $G_{2}$.

Lemma 2.10

[13]* Let $G$ be a graph of order $n$. $\gamma(G)=\frac{n}{2}$ if and only if the components of $G$ are the cycle $C_{4}$ or the corona $H\circ K_{1}$ for any connected graph $H$.*

Denote by $C_{3,\,k}^{*}$ the graph obtained by attaching a $C_{3}$ to an end vertex of a path of length $k$ and attaching $n-3-k$ pendant vertices to the other end vertex of this path.

Lemma 2.11

[17]* Among all the nonbipartite graphs with both order $n\geq 4$ and domination number $\gamma\leq\frac{n+1}{3}$, we have*

(i) if $n=3\gamma-1$, $3\gamma$, $3\gamma+1$, then the graph with the minimal least $Q$-eigenvalue attains uniquely at $C_{3,\,n-4}^{*}$;

(ii) if $n\geq 3\gamma+2$, then the graph with the minimal least $Q$-eigenvalue attains uniquely at $C_{3,\,3\gamma-3}^{*}$.

Lemma 2.12

[18]* Among all nonbipartite unicyclic graphs of order $n$, and with both domination number $\gamma$ and girth $g$ ($g\leq n-1$), the minimum $q_{min}$ attains at a $\mathcal{F}_{g,l}$-graph $G$ for some $l$. Moreover, for this graph $G$, suppose that $X=(x_{1}$, $x_{2}$, $x_{3}$, $\ldots$, $x_{n})^{T}$ is a unit eigenvector corresponding to $q_{min}(G)$. Then we have that $|x_{g}|>0$, and $|x_{g+l-1}|=\max\{|x_{i}|\mid v_{i}$ is a $p$-dominator$\,\}$.*

In $\mathcal{H}^{k}_{2}$, for $j=1$, $2$, $\ldots$, $k$, suppose $v_{\tau_{\varepsilon-2-k+j}}$ is the pendant vertex attached to vertex $v_{\varepsilon-2-k+j}$. Suppose $v_{\omega_{1}}$, $v_{\omega_{2}}$, $\ldots$, $v_{\omega_{s}}$ are the pendant vertices attached to vertex $v_{\varepsilon-1}$. If $s\geq 2$, let $\mathcal{H}^{k}_{3}=\mathcal{H}_{2}^{k}-v_{\varepsilon-1-k}v_{\tau_{\varepsilon-1-k}}+v_{\varepsilon-1}v_{\tau_{\varepsilon-1-k}}-\sum\limits_{j=2}^{s}v_{\varepsilon-1}v_{\omega_{j}}+\sum\limits_{j=2}^{s}v_{\omega_{1}}v_{\omega_{j}}$. Let $\mathcal{H}^{k-1}_{4}=\mathcal{H}^{k}_{2}-v_{\varepsilon-1-k}v_{\tau_{\varepsilon-1-k}}+v_{\varepsilon-1}v_{\tau_{\varepsilon-1-k}}$, $\mathcal{H}^{k-2}_{5}=\mathcal{H}^{k-1}_{4}-v_{\varepsilon-k}v_{\tau_{\varepsilon-k}}+v_{\varepsilon-1}v_{\tau_{\varepsilon-k}}$.

Lemma 2.13

[18**]****

$\mathrm{(i)}$* $\gamma(\mathcal{H}^{k}_{1})\leq\gamma(\mathcal{H}^{k}_{2})$.*

$\mathrm{(ii)}$* If $\varepsilon-k-1\leq 2$, then $\gamma(\mathcal{H}^{k}_{2})=k+1$ and $\gamma(\mathcal{H}^{k-1}_{4})=\gamma(\mathcal{H}^{k}_{2})-1$;*

$\mathrm{(iii)}$* If $\varepsilon-k-1\geq 3$, then $\gamma(\mathcal{H}^{k}_{2})=\lceil\frac{\varepsilon-k-4}{3}\rceil+k+1$;*

$\mathrm{(iv)}$* $\gamma(\mathcal{H}^{k}_{2})\leq\gamma(\mathcal{H}^{k}_{3})$;*

$\mathrm{(v)}$* If $\varepsilon-k-1\geq 3$, $\frac{\varepsilon-k-4}{3}\neq t$ where $t$ is a nonnegative integral number, then $\gamma(\mathcal{H}^{k-1}_{4})=\gamma(\mathcal{H}^{k}_{2})-1$;*

$\mathrm{(vi)}$* If $\varepsilon-k-1\geq 3$, $\frac{\varepsilon-k-4}{3}=t$ where $t$ is a nonnegative integral number, $\gamma(\mathcal{H}^{k-1}_{4})=\gamma(\mathcal{H}^{k}_{2})$, $\gamma(\mathcal{H}^{k-2}_{5})=\gamma(\mathcal{H}^{k}_{2})-1$.*

Lemma 2.14

[18**]****

$\mathrm{(i)}$* $\gamma(\mathscr{H}_{3,0})=1$;*

$\mathrm{(ii)}$* If $\alpha\geq 1$ and $n-2\alpha\leq 2$, then $\gamma(\mathscr{H}_{3,\alpha})=\alpha$;*

$\mathrm{(iii)}$* If $\alpha\geq 1$ and $n-2\alpha\geq 3$, then $\gamma(\mathscr{H}_{3,\alpha})=\lceil\frac{n-2\alpha-2}{3}\rceil+\alpha$.*

3 Domination number and the structure of a graph

Let $G^{\ast}$ be a sunlike graph of order $n$ and with both girth $g$ and $k$ $p$-dominators $v_{1}$, $v_{2}$, $\ldots$, $v_{k}$ on $\mathbb{C}$.

Lemma 3.1

Let $G$ be a sunlike graph of order $n$ and with both girth $g$ and $k$ $p$-dominators on $\mathbb{C}$. Then $\gamma(G)\leq\gamma(G^{\ast})$, where $\gamma(G^{\ast})=k+\lceil\frac{g-k-2}{3}\rceil$.

**Proof. ** Suppose $v_{i_{1}}$, $v_{i_{2}}$, $\ldots$, $v_{i_{k}}$ are the $k$ $p$-dominators on $\mathbb{C}$ in $G$, where $1\leq i_{1}<i_{2}<\cdots<i_{k}\leq g$. Suppose that there exists some $1\leq z\leq k$ such that $i_{z+1}-i_{z}\geq 2$, where if $z=k$, we let $i_{k+1}=i_{1}$ and $i_{k+1}-i_{k}=i_{1}+g-i_{k}$. Let $H=G-\sum^{i_{z+1}-1}_{s=i_{z}+1}v_{s}$.

Assertion 1 If $i_{z+1}-i_{z}\leq 3$, then $\gamma(H)=\gamma(G)$. By Lemma 2.8, there is a minimal dominating set $D$ of $G$ which contains all the $k$ $p$-dominators but no any pendant vertex. Thus both $v_{i_{z+1}}$ and $v_{i_{z}}$ are in $D$. Note the minimality of $D$ and $2\leq i_{z+1}-i_{z}\leq 3$. Then $D\cap\{v_{i_{z}+1}\}=\emptyset$ if $i_{z+1}-i_{z}=2$; $D\cap\{v_{i_{z}+1},v_{i_{z+1}-1}\}=\emptyset$ if $i_{z+1}-i_{z}=3$. Thus $D$ is also a dominating set of $H$. This implies that $\gamma(H)\leq\gamma(G)$. Note that for $H$, by Lemma 2.8, there is a minimal dominating set $D^{\prime}$ which contains all the $k$ $p$-dominators but no any pendant vertex. Thus both $v_{i_{z+1}}$ and $v_{i_{z}}$ are in $D^{\prime}$. Then $v_{i_{z}+1}$ is dominated by $D^{\prime}$ if $i_{z+1}-i_{z}=2$; $v_{i_{z}+1}$, $v_{i_{z+1}-1}$ are is dominated by $D^{\prime}$ if $i_{z+1}-i_{z}=3$. Consequently, $D^{\prime}$ is also a dominating set of $G$. This implies that $\gamma(G)\leq\gamma(H)$. As a result, it follows that $\gamma(H)=\gamma(G)$. And then our assertion holds.

Assertion 2 If $i_{z+1}-i_{z}\geq 4$, then $\gamma(G)=\gamma(H)+\gamma(P_{i_{z},i_{z+1}})$ where $P_{i_{z},i_{z+1}}=v_{i_{z}+2}v_{i_{z}+3}\cdots$ $v_{i_{z+1}-2}$. By Lemma 2.8, there is a minimal dominating set $D$ of $G$ which contains all the $k$ $p$-dominators but no any pendant vertex. Thus both $v_{i_{z+1}}$ and $v_{i_{z}}$ are in $D$. We claim that at most one of $v_{i_{z}+1}$, $v_{i_{z}+2}$ is in $D$. Otherwise, suppose that both $v_{i_{z}+1}$ and $v_{i_{z}+2}$ are in $D$. Then $D\setminus\{v_{i_{z}+1}\}$ is also a dominating set of $G$, which contradicts the minimality of $D$. Consequently, our claim holds. Similarly, we get that at most one of $v_{i_{z+1}-2}$, $v_{i_{z+1}-1}$ is in $D$. Thus we let $D^{\circ}=((D\cup\{v_{i_{z}+2},v_{i_{z+1}-2}\})\setminus\{v_{i_{z}+1},v_{i_{z+1}-1}\})\cap V(P_{i_{z},i_{z+1}})$ if $v_{i_{z}+1}\in D$, $v_{i_{z+1}-1}\in D$; let $D^{\circ}=((D\cup\{v_{i_{z}+2}\})\setminus\{v_{i_{z}+1}\})\cap V(P_{i_{z},i_{z+1}})$ if $v_{i_{z}+1}\in D$ and $v_{i_{z+1}-1}\notin D$; let $D^{\circ}=((D\cup\{v_{i_{z+1}-2}\})\setminus\{v_{i_{z+1}-1}\})\cap V(P_{i_{z},i_{z+1}})$ if $v_{i_{z}+1}\notin D$ and $v_{i_{z+1}-1}\in D$; let $D^{\circ}=(D\cap V(P_{i_{z},i_{z+1}})$ if $v_{i_{z}+1}\notin D$ and $v_{i_{z+1}-1}\notin D$. Note that $D^{\ast}=D\setminus(V(P_{i_{z},i_{z+1}})\cup\{v_{i_{z}+1}$, $v_{i_{z+1}-1}\})$ is a dominating set of $H$, $D^{\circ}\cup D^{\ast}$ is a dominating set of $G$ with cardinality $\gamma(G)$, and note that $D^{\circ}$ is a dominating set of $P_{i_{z},i_{z+1}}$. Thus $\gamma(P_{i_{z},i_{z+1}})\leq\parallel D^{\circ}\parallel$. Note that both $v_{i_{z+1}-1}$ and $v_{i_{z}+1}$ are dominated by $D^{\ast}$. Consequently, for any minimal dominating set $B$ of $P_{i_{z},i_{z+1}}$, then $B\cup D^{\ast}$ is also a dominating set of $G$. Note that $\parallel B\parallel=\gamma(P_{i_{z},i_{z+1}})\leq\parallel D^{\circ}\parallel$. As a result, $\parallel B\cup D^{\ast}\parallel\leq\parallel D\parallel=\gamma(G)$. Note that the minimality of $D$. Then $\parallel D^{\circ}\parallel=\parallel B\parallel=\gamma(P_{i_{z},i_{z+1}})$, and then it follows that $\gamma(G)=\gamma(H)+\gamma(P_{i_{z},i_{z+1}})$.

Denote by $\tau_{i_{j},i_{j+1}}$ the dominating index where we let $i_{k+1}=i_{1}$ if $i=k$. Let $\tau_{i_{j},i_{j+1}}=0$ if $i_{j+1}-i_{j}\leq 3$; let $\tau_{i_{j},i_{j+1}}=\gamma(P_{i_{j},i_{j+1}})$ if $i_{j+1}-i_{j}\geq 4$. Thus from Assertion 1, Assertion 2 and Lemma 2.8, we get that $\gamma(G)=k+\sum^{k}_{i=1}\tau_{i_{j},i_{j+1}}$. By Lemma 2.9, it follows that $\tau_{i_{j},i_{j+1}}=\gamma(P_{i_{j},i_{j+1}})=\lceil\frac{i_{j+1}-i_{j}-3}{3}\rceil$ if $i_{j+1}-i_{j}\geq 4$. Note that for any two nonnegative integers $x$ and $y$, we have $\lceil\frac{x}{3}\rceil+\lceil\frac{y}{3}\rceil\leq\lceil\frac{x+y}{3}\rceil$. Then

[TABLE]

Thus $\gamma(G)\leq k+\lceil\frac{g-k-2}{3}\rceil$. Noting that by Assertion1 and Assertion 2, we have $\gamma(G^{\ast})=k+\lceil\frac{g-k-2}{3}\rceil$. Then the result follows as desired. This completes the proof.      $\Box$

Theorem 3.2

Suppose that $\mathcal{G}$ is a nonbipartite $\mathcal{F}_{g,l}$-graph with $\gamma(\mathcal{G})=\frac{n-1}{2}$, $g\geq 5$ and order $n\geq g+1$, and suppose there are exactly $f$ vertices of the unique cycle $\mathbb{C}$ such that none of them is $p$-dominator. Then we get

$\mathrm{(i)}$* if $f=g$, then $g=5$;*

$\mathrm{(ii)}$* if $f\neq g$, then $f\leq 3$ and $f\neq 2$;*

$\mathrm{(iii)}$* if $f=3$, then the three vertices are consecutive on $\mathbb{C}$, i.e., they are $v_{i-1}$, $v_{i}$, $v_{i+1}$ for some $1\leq i<g$, and each in $(V(\mathbb{C})\setminus\{v_{i-1}$, $v_{i}$, $v_{i+1}\})\cup V(\mathbb{P}-v_{g+l})$ is a $p$-dominator (if $i=1$, then $v_{i-1}=v_{g}$).*

**Proof. ** Denote by $A$ the set of vertices of $\mathbb{C}$ and the pendant vertices attached to $\mathbb{C}$. Let $\parallel A\parallel=z$, and let $A^{{}^{\prime}}=V(\mathcal{G})\setminus A$. Then $\gamma(\mathcal{G})\leq\gamma(\mathcal{G}[A])+\gamma(\mathcal{G}[A^{{}^{\prime}}])$. Note that $A^{{}^{\prime}}=\emptyset$, or $\mathcal{G}[A^{{}^{\prime}}]$ is connected with at least $2$ vertices. Suppose $f\geq 4$.

(i) $f=g$. Then $z-f=0$. This means that there is no $p$-dominator on $\mathbb{C}$. So, $\mathcal{G}[A^{{}^{\prime}}]$ is connected with at least $2$ vertices. Thus, if $f\geq 9$, by Lemma 2.9, then $\gamma(\mathcal{G})\leq\lceil\frac{f}{3}\rceil+\gamma(\mathcal{G}[A^{{}^{\prime}}])\leq\frac{n-f}{2}+\frac{f+2}{3}<\frac{n-1}{2}$. Therefore $f\leq 7$.

Note that $g$ is odd and $g=f$ now. Thus if $\gamma(\mathcal{G}[A^{{}^{\prime}}])<\frac{n-f}{2}$, then $\gamma(\mathcal{G})\leq\lceil\frac{f}{3}\rceil+\gamma(\mathcal{G}[A^{{}^{\prime}}])<\frac{n-1}{2}$. Hence, it follows that $\gamma(\mathcal{G}[A^{{}^{\prime}}])=\frac{n-f}{2}$. Combined with Lemma 2.10, it follows that $\mathcal{G}[A^{{}^{\prime}}]=P_{\frac{n-f}{2}}\circ K_{1}$. Here, suppose $P_{\frac{n-f}{2}}=v_{a_{1}}v_{a_{2}}\cdots v_{a_{t}}$ with $t=\frac{n-f}{2}$, and suppose $v_{\tau_{1}}$ is the unique pendant vertex attached to $v_{a_{1}}$. By Lemma 2.8, $V(P_{\frac{n-f}{2}})$ is a minimal dominating set of $\mathcal{G}[A^{{}^{\prime}}]$.

Assume that $f=7$. Note that $\mathcal{G}$ is a $\mathcal{F}_{g,l}$-graph. If $\mathcal{G}=\mathbb{C}+v_{g}v_{a_{1}}+\mathcal{G}[A^{{}^{\prime}}]$, then $V(P_{\frac{n-f}{2}})\cup\{v_{2},v_{5}\}$ is a dominating set of $\mathcal{G}$; if $\mathcal{G}=\mathbb{C}+v_{g}v_{\tau_{1}}+\mathcal{G}[A^{{}^{\prime}}]$, then $(V(P_{\frac{n-f}{2}})\backslash\{v_{a_{1}}\})\cup\{v_{2},v_{5},v_{\tau_{1}}\}$ is a dominating set of $\mathcal{G}$. This implies that $\gamma(\mathcal{G})\leq\frac{n-7}{2}+2<\frac{n-1}{2}$ which contradicts $\gamma(\mathcal{G})=\frac{n-1}{2}$. Thus, it follows that $g=5$.

(ii) $f\neq g$. Note that there is no the case that $z-f=1$. Then $z-f\geq 2$. By Lemma 3.1, $\gamma(\mathcal{G}[A])\leq\gamma(\mathcal{G}^{\ast}[A])=g-f+\lceil\frac{f-2}{3}\rceil\leq\frac{z-f}{2}+\lceil\frac{f-2}{3}\rceil$, where $\mathcal{G}^{\ast}[A]$ is a sunlike graph with vertex set $A$, $\mathbb{C}$ contained in it and $g-f$ $p$-dominators $v_{1}$, $v_{2}$, $\ldots$, $v_{g-f}$ (defined as $\mathcal{G}^{\ast}$ in Lemma 3.1). Thus, if $f\geq 4$, then $\gamma(\mathcal{G})\leq\frac{z-f}{2}+\lceil\frac{f-2}{3}\rceil+\gamma(\mathcal{G}[A^{{}^{\prime}}])\leq\frac{n-f}{2}+\lceil\frac{f-2}{3}\rceil\leq\frac{n-f}{2}+\frac{f}{3}<\frac{n-1}{2}$. This contradicts that $\gamma(\mathcal{G})=\frac{n-1}{2}$. Consequently, $f\leq 3$.

Suppose $f=2$ and suppose that $v_{j}$, $v_{k}$ of $\mathbb{C}$ are the exact $2$ vertices such that neither of them is $p$-dominator. Note that by Lemma 2.8, there is a minimal dominating set $D$ of $\mathcal{G}-v_{j}-v_{k}$ which contains all $p$-dominators but no any pendant vertex. Note that the vertices of $\mathbb{C}$ other than $v_{j}$, $v_{k}$ are all $p$-dominators in both $\mathcal{G}-v_{j}-v_{k}$ and $\mathcal{G}$. Thus, each of $v_{j}$, $v_{k}$ is adjacent to at least one $p$-dominator on $\mathbb{C}$. So, $D$ is also a dominating set of $\mathcal{G}$. Note that there is no isolated vertex in $\mathcal{G}-v_{j}-v_{k}$. Then $\gamma(\mathcal{G}-v_{j}-v_{k})\leq\frac{n-2}{2}$, and then $\gamma(\mathcal{G})\leq\frac{n-2}{2}$, which contradicts $\gamma(\mathcal{G})=\frac{n-1}{2}$. Then (ii) follows.

(iii) Suppose $v_{a}$, $v_{b}$, $v_{c}$ are the exact $3$ vertices of $\mathbb{C}$ such that none of them is $p$-dominator. If the $3$ vertices $v_{a}$, $v_{b}$, $v_{c}$ are not consecutive, then each of them can be dominated by its adjacent $p$-dominator. Note that by Lemma 2.8, there are a minimal dominating set $D$ of $\mathcal{G}-v_{a}-v_{b}-v_{c}$ which contains all $p$-dominators but no any pendant vertex. Thus such $D$ is also a dominating set of $\mathcal{G}$. Note that there is no isolated vertex in $\mathcal{G}-v_{a}-v_{b}-v_{c}$. So, $\gamma(\mathcal{G})\leq\parallel D\parallel=\gamma(\mathcal{G}-v_{a}-v_{b}-v_{c})\leq\frac{n-3}{2}$, which contradicts $\gamma(\mathcal{G})=\frac{n-1}{2}$. Therefore, the $3$ vertices $v_{a}$, $v_{b}$, $v_{c}$ are consecutive.

Suppose that the $3$ vertices are $v_{i-1}$, $v_{i}$, $v_{i+1}$ for some $1\leq i\leq g$ (here, if $i=g$, we let $v_{i+1}=v_{1}$; if $i=1$, we let $v_{i-1}=v_{g}$). Let $H=\mathcal{G}-v_{i-1}-v_{i}-v_{i+1}$. Note that there is no isolated vertex in $H$. Thus, $\gamma(H)\leq\frac{n-3}{2}$. Next, we claim that $\gamma(H)=\frac{n-3}{2}$.

Claim 1 $\gamma(H)=\frac{n-3}{2}$. Otherwise, suppose $\gamma(H)<\frac{n-3}{2}$, and suppose $D$ is a minimal dominating set of $H$. Then $D\cup\{v_{i}\}$ is a dominating set $D$ of $\mathcal{G}$. Thus, $\mathcal{G}<1+\frac{n-3}{2}<\frac{n-1}{2}$, which contradicts $\gamma(\mathcal{G})=\frac{n-1}{2}$. Then the claim holds.

By Lemma 2.10, $H=\mathcal{L}\circ K_{1}$ for some acyclic graph $\mathcal{L}$ of order $\frac{n-3}{2}$.

Claim 2 For any minimal dominating set $D$ of $H$, in $\mathcal{G}$, at least one of $v_{i-1}$, $v_{i}$, $v_{i+1}$ can not be dominated by $D$. Otherwise, $D$ is a dominating set of $\mathcal{G}$ too. Hence, $\gamma(\mathcal{G})\leq\frac{n-3}{2}$, which contradicts $\gamma(\mathcal{G})=\frac{n-1}{2}$. Then the claim holds.

If $i=g$, then let $H=H_{1}\cup H_{2}$, where $H_{1}=\mathcal{G}[A]-v_{g-1}-v_{g}-v_{1}$, $H_{2}=\mathcal{G}[A^{{}^{\prime}}]=P_{\frac{n-z}{2}}\circ K_{1}$ (if $n=z$, then $H_{2}$ is empty). Here, suppose $P_{\frac{n-z}{2}}=v_{a_{1}}v_{a_{2}}\cdots v_{a_{t}}$ with $t=\frac{n-z}{2}$, and suppose $v_{\tau_{1}}$ is the unique pendant vertex attached to $v_{a_{1}}$. Thus there are two possible cases for $G$, i.e., $\mathcal{G}=\mathcal{G}[A]+v_{g}v_{a_{1}}+H_{2}$ or $\mathcal{G}=\mathcal{G}[A]+v_{g}v_{\tau_{1}}+H_{2}$. Let $\mathcal{Z}=(\mathbb{C}\setminus\{v_{g-1},v_{g},v_{1}\})\cup V(P_{\frac{n-z}{2}})$. Note that the vertices in $\mathcal{Z}$ are all $p$-dominators in $\mathcal{G}$. If $\mathcal{G}=\mathcal{G}[A]+v_{g}v_{a_{1}}+H_{2}$, then $\mathcal{Z}$ is also a dominating set of $\mathcal{G}$; if $\mathcal{G}=\mathcal{G}[A]+v_{g}v_{\tau_{1}}+H_{2}$, then $(\mathcal{Z}\setminus\{v_{a_{1}}\})\cup\{v_{\tau_{1}}\}$ is a dominating set of $\mathcal{G}$. Thus it follows that $\gamma(\mathcal{G})\leq\frac{n-3}{2}<\frac{n-1}{2}$ which contradicts $\gamma(\mathcal{G})=\frac{n-1}{2}$. This implies $i\neq g$.

If $i\neq 1,g-1$, then $H$ is connected. Let $\mathcal{Z}=(V(\mathbb{C})\setminus\{v_{i-1}$, $v_{i}$, $v_{i+1}\})\cup V(\mathbb{P}-v_{g+l})$, where $\mathbb{P}=v_{g}v_{g+1}\cdots v_{g+l}$. Then each vertex in $\mathcal{Z}$ is a $p$-dominator in $\mathcal{G}$.

If $i=1$, then let $H=H_{1}\cup H_{2}$, where $H_{1}=\mathcal{G}[A]-v_{g}-v_{1}-v_{2}$, $H_{2}=\mathcal{G}[A^{{}^{\prime}}]=P_{\frac{n-z}{2}}\circ K_{1}$ (if $n=z$, then $H_{2}$ is empty). Here, suppose $P_{\frac{n-z}{2}}=v_{a_{1}}v_{a_{2}}\cdots v_{a_{t}}$ with $t=\frac{n-z}{2}$, and suppose $v_{\tau_{1}}$ is the unique pendant vertex attached to $v_{a_{1}}$. Thus there are two possible cases for $G$, i.e., $\mathcal{G}=\mathcal{G}[A]+v_{g}v_{a_{1}}+H_{2}$ or $\mathcal{G}=\mathcal{G}[A]+v_{g}v_{\tau_{1}}+H_{2}$. We say that $\mathcal{G}\neq\mathcal{G}[A]+v_{g}v_{\tau_{1}}+H_{2}$. Otherwise, suppose $\mathcal{G}=\mathcal{G}[A]+v_{g}v_{\tau_{1}}+H_{2}$. Note that $n-z$ is even now and $\mathcal{G}-\{v_{2},v_{1},v_{g},v_{a_{1}},v_{\tau_{1}}\}$ has no isolated vertex. Then for $\mathcal{G}-\{v_{2},v_{1},v_{g},v_{a_{1}},v_{\tau_{1}}\}$, it has a dominating set $\mathbb{D}$ with $\parallel\mathbb{D}\parallel\leq\frac{n-5}{2}$. Then $\mathbb{D}\cup\{v_{1},v_{\tau_{1}}\}$ is a dominating set of $\mathcal{G}$, which contradicts $\gamma(\mathcal{G})=\frac{n-1}{2}$. This implies that $\mathcal{G}=\mathcal{G}[A]+v_{g}v_{a_{1}}+H_{2}$. It follows that each one in $(V(\mathbb{C})\setminus\{v_{g}$, $v_{1}$, $v_{2}\})\cup V(\mathbb{P}-v_{g+l})$ is a $p$-dominator. Similarly, for $i=g-1$, we get that each one in $(V(\mathbb{C})\setminus\{v_{g-2}$, $v_{g-1}$, $v_{g}\})\cup V(\mathbb{P}-v_{g+l})$ is a $p$-dominator. Then (iii) follows.      $\Box$

4 The $q_{min}$ among uncyclic graphs

Lemma 4.1

[18]* Let $G$ be a nonbipartite unicyclic graph of order $n$ and with the odd cycle $\mathcal{C}=v_{1}v_{2}\cdots v_{g}v_{1}$ in it. There is a unit eigenvector $X=(\,x_{1}$, $x_{2}$, $\ldots$, $x_{g}$, $x_{g+1}$, $x_{g+2}$, $\ldots$, $x_{n-1}$, $x_{n}\,)^{T}$ corresponding to $q_{min}(G)$, in which suppose $|x_{1}|=\min\{|x_{1}|$, $|x_{2}|$, $\ldots$, $|x_{g}|\}$, $|x_{s}|=\max\{|x_{1}|$, $|x_{2}|$, $\ldots$, $|x_{g}|\}$ where $s\geq 2$, satisfying that*

$\mathrm{(i)}$* $|x_{1}|<|x_{s}|$;*

$\mathrm{(ii)}$* $|x_{1}|=0$ if and only if $x_{g}=-x_{2}\neq 0$; if $|x_{1}|=0$ and $x_{i}x_{i+1}\neq 0$ for some $1\leq i\leq g-1$, then $x_{i}x_{i+1}<0$; moreover, if $x_{j}\neq 0$, then $sgn(x_{j})=(-1)^{d_{istH}(v_{1},v_{j})}$ where $H=G-v_{1}v_{g}$.*

$\mathrm{(iii)}$* if $|x_{1}|>0$, then*

$\mathrm{(1)}$* if $3\leq s\leq g-1$, then $|x_{2}|<\cdots<|x_{s-2}|<|x_{s-1}|\leq|x_{s}|$ and $|x_{g}|<|x_{g-1}|<\cdots<|x_{s+2}|<|x_{s+1}|\leq|x_{s}|$;*

$\mathrm{(2)}$* if $|x_{2}|>|x_{g}|$, then $x_{1}x_{g}>0$; for $1\leq i\leq g-1$, $x_{i}x_{i+1}<0$; $|x_{1}|\leq|x_{g}|$;*

$\mathrm{(3)}$* if $|x_{2}|<|x_{g}|$, then $x_{1}x_{2}>0$; for $2\leq i\leq g-1$, $x_{i}x_{i+1}<0$; $x_{g}x_{1}<0$; $|x_{1}|\leq|x_{2}|$;*

$\mathrm{(4)}$* if $|x_{2}|=|x_{g}|$, then $|x_{1}|\leq|x_{2}|$, and exactly one of $x_{1}x_{g}>0$ and $x_{1}x_{2}>0$ holds, where*

$\mathrm{(4.1)}$* if $x_{1}x_{g}>0$, then for $1\leq i\leq g-1$, $x_{i}x_{i+1}<0$;*

$\mathrm{(4.2)}$* if $x_{1}x_{2}>0$, then $x_{i}x_{i+1}<0$ for $2\leq i\leq g-1$ and $x_{g}x_{1}<0$;*

$\mathrm{(5)}$* at least one of $|x_{s+1}|$ and $|x_{s-1}|$ is less than $|x_{s}|$.*

Lemma 4.2

[18]* If $\mathcal{G}$ is a nonbipartite $\mathcal{F}^{\circ}_{g,l}$-graph with $g\geq 5$, $n\geq g+1$, then there is a graph $\mathbb{H}$ with girth $3$ and order $n$ such that $\gamma(\mathcal{G})\leq\gamma(\mathbb{H})$ and $q_{min}(\mathbb{H})<q_{min}(\mathcal{G})$.*

Lemma 4.3

[18]* Suppose that $G$ is a nonbipartite $\mathcal{F}_{3,l}$-graph of order $n$ where $\mathbb{C}=v_{1}v_{2}v_{3}v_{1}$. $X=(\,x_{1}$, $x_{2}$, $\ldots$, $x_{n}\,)^{T}$ is a unit eigenvector corresponding to $q_{min}(G)$. Then $|x_{3}|=\max\{|x_{1}|$, $|x_{2}|$, $|x_{3}|\}$.*

Theorem 4.4

Among all nonbipartite unicyclic graphs of order $n\geq 5$ with girth $3$ and domination number at least $\frac{n+1}{3}<\gamma\leq\frac{n}{2}$, if $\gamma=\frac{n-1}{2}$, the $q_{min}$ attains the minimum uniquely at $\mathscr{H}_{3,\frac{n-3}{2}}$.

**Proof. ** The result follows from Lemmas 2.4, 2.12, 2.13, 4.3 and Theorem 3.2      $\Box$

Let $\mathcal{K}=\{G|\,\ G\ \mathrm{be}\ \mathrm{a}\ \mathrm{nonbipartite}\ \mathcal{F}^{\circ}_{g,l}$-graph of order $n\geq 4$ and domination number at least $\frac{n+1}{3}<\gamma\leq\frac{n}{2}$, where $g$ is any odd number at least $3$ and $l$ is any positive integral number$\}$ and $q_{\mathcal{K}}=\min\{q_{min}(G)|\,\ G\in\mathcal{K}\}$.

Lemma 4.5

[18**]****

$\mathrm{(i)}$* If $n=4$, the $q_{\mathcal{K}}$ attains uniquely at $\mathscr{H}_{3,1}$;*

$\mathrm{(ii)}$* If $n\geq 5$ and $n-2\gamma\geq 2$, then the least $q_{\mathcal{K}}>q_{min}(\mathscr{H}_{3,\alpha})$ where $\alpha\leq\frac{n-3}{2}$ is the least integer such that $\lceil\frac{n-2\alpha-2}{3}\rceil+\alpha=\gamma$.*

Lemma 4.6

For a nonbipartite $\mathcal{F}_{g,l}$-graph graph $G$ of order $n\geq 5$ and with $g=5$, there exists a graph $\mathbb{H}$ such that $g(\mathbb{H})=3$, $\gamma(G)\leq\gamma(\mathbb{H})$ and $q_{min}(\mathbb{H})<q_{min}(G)$.

**Proof. ** If $n=5$, then $G=C_{5}$. And then the result follows from Lemma 2.11. Next we consider the case that $n\geq 6$. By Lemma 2.6, we get that $q_{min}(G)<1$.

Case 1 There is no $p$-dominator on $\mathbb{C}$. Then $G$ is like $G_{1}$ (see $G_{1}$ in Fig. 4.1). By Lemma 2.5, there is a unit eigenvector $X=(\,x_{1}$, $x_{2}$, $\ldots$, $x_{k}$, $x_{k+1}$, $x_{k+2}$, $\ldots$, $x_{n-1}$, $x_{n}\,)^{T}$ corresponding to $q_{min}(G)$ such that $|x_{5}|=\max\{|x_{1}|$, $|x_{2}|$, $|x_{3}|$, $|x_{4}|$, $|x_{5}|\}>0$, and $x_{1}=x_{4}$, $x_{2}=x_{3}$. By Lemma 4.1, we get that $|x_{2}|>0$, $|x_{2}|<|x_{1}|$ and $x_{2}x_{1}<0$. Let $\mathbb{H}=G-v_{3}v_{4}+v_{3}v_{1}$. By Lemma 2.4, we get that $q_{min}(\mathbb{H})<q_{min}(G)$. Let $B_{1}=\mathbb{H}[v_{1}$, $v_{2}$, $v_{3}]$, $B_{2}=\mathbb{H}-\{v_{1}$, $v_{2}$, $v_{3}\}$. As Lemma 3.1, we can get a minimal dominating set $D$ of $\mathbb{H}$, which contains all $p$-dominators but no any pendant vertex and no $v_{1}$, such that $D=\{v_{2}\}\cup D_{2}$, where $\{v_{2}\}$ is a dominating set of $B_{1}$, $D_{2}$ is a dominating set of $B_{2}$. Note that $D$ is also a dominating set of $G$. So, $\gamma(G)\leq\gamma(\mathbb{H})$.

v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{5+s}$$G_{1}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$G_{2}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{5+s}$$G_{3}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{s+5}$$G_{4}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$G_{5}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$G_{6}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{s+5}$$G_{7}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{s+5}$$G_{8}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{s+5}$$G_{9}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$G_{10}$$v_{s+5}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$G_{11}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$G_{12}$$G_{13}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{5+s}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{5+s}$$G_{14}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$G_{15}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$G_{16}$$v_{s+5}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$G_{19}Fig. 4.1. G_{1}-G_{19}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$G_{17}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$G_{18}

Case 2 There is only $1$ $p$-dominator on $\mathbb{C}$ (see $G_{2}-G_{4}$ in Fig. 4.1).

Subcase 2.1 For $G_{2}$, let $\mathbb{H}=G_{2}-v_{3}v_{4}+v_{3}v_{1}$. As Case 1, it is proved that $\gamma(G_{2})\leq\gamma(\mathbb{H})$ and $q_{min}(\mathbb{H})<q_{min}(G_{2})$.

Subcase 2.2 For $G_{3}$, suppose $X=(\,x_{1}$, $x_{2}$, $\ldots$, $x_{n-1}$, $x_{n}\,)^{T}$ is a unit eigenvector corresponding to $q_{min}(G_{3})$.

Claim $|x_{4}|>|x_{1}|$, $|x_{5}|>|x_{3}|$. Denote by $v_{k}$ the pendant vertex attached to $v_{4}$. Suppose $0<|x_{4}|\leq|x_{1}|$. Let $G^{{}^{\prime}}_{3}=G_{3}-v_{4}v_{k}+v_{1}v_{k}$. By Lemma 2.4, then $q_{min}(G^{{}^{\prime}}_{3})<q_{min}(G_{3})$. This is a contradiction because $G^{{}^{\prime}}_{3}\cong G_{3}$. Suppose $|x_{4}|=|x_{1}|=0$. By Lemma 4.1, we get that $x_{2}\neq 0$, $x_{3}\neq 0$. By $q_{min}(G_{3})x_{2}=2x_{2}+x_{3}$, $q_{min}(G_{3})x_{3}=2x_{3}+x_{2}$, we get $x^{2}_{2}=x^{2}_{3}$. Suppose $x_{2}>0$. Then we get $q_{min}(G_{3})x_{2}=2x_{2}+x_{3}\geq x_{2}$. This means that $q_{min}(G_{3})\geq 1$ which contradicts $q_{min}(G_{3})<1$. Thus, $|x_{4}|>|x_{1}|$. Similarly, we get $|x_{5}|>|x_{3}|$. Then the claim holds.

Suppose $|x_{1}|=\min\{|x_{1}|$, $|x_{2}|$, $|x_{3}|\}$ and $x_{1}\geq 0$. If $|x_{2}|>|x_{5}|$, by Lemma 4.1, suppose $x_{1}x_{5}\geq 0$. Let $H=G_{3}-v_{1}v_{5}$. Also by Lemma 4.1, suppose for any $j\neq 1,5$, $\mathrm{sgn}x_{j}=(-1)^{d_{istH}(v_{j},v_{1})}$. Let $\mathbb{H}=G_{3}-v_{1}v_{5}+v_{3}v_{1}$. Because $|x_{5}|>|x_{3}|$, it follows that $q_{min}(\mathbb{H})\leq X^{T}Q(\mathbb{H})X<X^{T}Q(G_{3})X=q_{min}(G_{3})$. Let $B_{1}=\mathbb{H}[v_{1},v_{2}]$, $B_{2}=\mathbb{H}-\{v_{1},v_{2}\}$. As Lemma 3.1, we can get a minimal dominating set $D$ of $\mathbb{H}$, which contains all $p$-dominators but no any pendant vertex and no $v_{3}$, such that $D=\{v_{1}\}\cup D_{2}$, where $D_{2}$ is a dominating set of $B_{2}$. Note that $D$ is also a dominating set of $G_{3}$. So, $\gamma(G_{3})\leq\gamma(\mathbb{H})$. If $|x_{2}|<|x_{5}|$, by Lemma 4.1, $x_{1}x_{2}\geq 0$. Let $H=G_{3}-v_{1}v_{2}$. Also by Lemma 4.1, suppose for any $j\neq 1,2$, $\mathrm{sgn}x_{j}=(-1)^{d_{istH}(v_{j},v_{1})}$. Let $\mathbb{H}=G_{3}-v_{1}v_{5}+v_{3}v_{1}$. Because $|x_{5}|>|x_{3}|$, it follows that $q_{min}(\mathbb{H})<q_{min}(G_{3})$ similarly. As the case that $|x_{2}|>|x_{5}|$, it is proved that $\gamma(G_{3})\leq\gamma(\mathbb{H})$. If $|x_{2}|=|x_{5}|$, by Lemma 4.1, without loss of generality, suppose $x_{1}x_{5}\geq 0$. Let $\mathbb{H}=G_{3}-v_{1}v_{5}+v_{3}v_{1}$. As the case that $|x_{2}|>|x_{5}|$, it is proved that $q_{min}(\mathbb{H})<q_{min}(G_{3})$, $\gamma(G_{3})\leq\gamma(\mathbb{H})$.

For the both cases that $|x_{2}|=\min\{|x_{1}|$, $|x_{2}|$, $|x_{3}|\}$ and $|x_{3}|=\min\{|x_{1}|$, $|x_{2}|$, $|x_{3}|\}$. As the case that $|x_{1}|=\min\{|x_{1}|$, $|x_{2}|$, $|x_{3}|\}$, it is proved that there exists a graph $\mathbb{H}$ such that $g(\mathbb{H})=3$, $\gamma(G_{3})\leq\gamma(\mathbb{H})$ and $q_{min}(\mathbb{H})<q_{min}(G_{3})$.

In a same way, for $G_{4}$, it is proved that there exists a graph $\mathbb{H}$ such that $g(\mathbb{H})=3$, $\gamma(G_{4})\leq\gamma(\mathbb{H})$ and $q_{min}(\mathbb{H})<q_{min}(G_{4})$.

And in a same way, for the cases that Case 3 there is exactly $2$ $p$-dominators on $\mathbb{C}$ (see $G_{5}-G_{10}$ in Fig. 4.1); Case 4 there is exactly $3$ $p$-dominators on $\mathbb{C}$ (see $G_{11}-G_{15}$ in Fig. 4.1); Case 5 there is exactly $4$ $p$-dominators on $\mathbb{C}$ (see $G_{16}-G_{18}$ in Fig. 4.1); Case 6 there is exactly $5$ $p$-dominators on $\mathbb{C}$ (see $G_{19}$ in Fig. 4.1), it is proved that the exists a a graph $\mathbb{H}$ such that $g(\mathbb{H})=3$, $\gamma(G)\leq\gamma(\mathbb{H})$ and $q_{min}(\mathbb{H})\leq q_{min}(G)$. Thus, the result follows as desired.      $\Box$

v_{2}$$v_{3}$$v_{4}$$v_{1}$$v_{\frac{n+5}{2}}$$v_{\frac{n+3}{2}}$$v_{\frac{n+1}{2}}Fig. 4.2. $\mathscr{H}_{3,\frac{n-3}{2}}$

Lemma 4.7

Let $G$ be a nonbipartite $\mathcal{F}_{g,l}$-graph of order $n$ for some $l$ and with domination number $\frac{n-1}{2}$. Then $q_{min}(G)\geq q_{min}(\mathscr{H}_{3,\frac{n-3}{2}})$ with equality if and only if $G\cong\mathscr{H}_{3,\frac{n-3}{2}}$ (see Fig. 4.2).

**Proof. ** Because $G$ is nonbipartite, $g$ is odd. If $G$ is a $\mathcal{F}^{\circ}_{g,l}$-graph, then the theorem follows from Lemma 4.5. If $g=3$, then the theorem follows from Theorem 4.4. For $g=5$, the theorem follows from Lemma 4.6. Next we consider the case that $G$ is not a $\mathcal{F}^{\circ}_{g,l}$-graph and suppose $g\geq 7$.

Let $X=(\,x_{1}$, $x_{2}$, $\ldots$, $x_{n}\,)^{T}$ is a unit eigenvector corresponding to $q_{min}(G)$. Suppose $x_{a}=\min\{|x_{1}|$, $|x_{2}|$, $\ldots$, $|x_{g}|\}$. Note that by Theorem 3.2, in $G$, there are at most $3$ consecutive vertices of $\mathbb{C}$ such that none of them is $p$-dominator, and there are $2$ cases as follows to consider.

Case 1 In $G$, there is exactly one vertex of $\mathbb{C}$ which is not $p$-dominator. Note that $G$ is not a $\mathcal{F}^{\circ}_{g,l}$-graph. Then $n\geq g+2$ and $v_{g}$ is the only one vertex which is not $p$-dominator on $\mathbb{C}$. By a same discussion in the proof of Lemma 4.3 (see [18]), it is proved that $x_{g}=\max\{|x_{1}|$, $|x_{2}|$, $\ldots$, $|x_{g-1}|$, $|x_{g}|\}$. Then we suppose $a\leq g-1$. By Lemma 4.1, if $a\leq g-3$, without loss of generality, suppose $x_{a+1}\leq x_{a-1}$, $x_{a+1}x_{a}\geq 0$, $|x_{a-1}|\geq|x_{a+2}|$. Let $G_{1}=G-v_{a}v_{a-1}+v_{a}v_{a+2}$ (if $|x_{a-1}|\leq|x_{a+2}|$ and $a\geq 2$, let $G_{1}=G-v_{a+1}v_{a+2}+v_{a+1}v_{a-1}$; if $a=1$, let $G_{1}=G-v_{1}v_{g}+v_{1}v_{3}$). If $a=g-2$, suppose $|x_{g-1}|\leq|x_{g-3}|$, $x_{g-1}x_{g-2}\geq 0$, and then let $G_{1}=G-v_{g-1}v_{g}+v_{g-1}v_{g-3}$. If $a=g-1$, because $|x_{g}|\geq|x_{g-2}|$, then suppose $x_{g-1}x_{g-2}\geq 0$. Let $G_{1}=G-v_{g-1}v_{g}+v_{g-1}v_{g-3}$. Note that $\gamma(G_{1})\leq\frac{n-1}{2}$. As the proof of Lemma 4.2, we get that $\gamma(G)\leq\gamma(G_{1})=\frac{n-1}{2}$, $q_{min}(G_{1})<q_{min}(G)$. Note that $g(G_{1})=3$. Then the theorem follows from Theorem 4.4.

Case 2 In $G$, there are exactly $3$ consecutive vertices of $\mathbb{C}$ such that each of them is not $p$-dominator. Note that $G$ is not a $\mathcal{F}^{\circ}_{g,l}$-graph. Combined with Theorem 3.2, the $3$ vertices of $\mathbb{C}$ such that each of them is not $p$-dominator are $v_{g-2}$, $v_{g-1}$, $v_{g}$ or $v_{g}$, $v_{1}$, $v_{2}$. Without loss of generality, we suppose the $3$ vertices are $v_{g-2}$, $v_{g-1}$, $v_{g}$. By Lemma 2.12, $|x_{g}|>0$. We say that $|x_{g}|>|x_{g-2}|$. Otherwise, suppose $|x_{g}|\leq|x_{g-2}|$. Let $G^{{}^{\prime}}=G-v_{g}v_{g+1}+v_{g+1}v_{g-2}$. Then by Lemma 2.4, $q_{min}(G^{{}^{\prime}})<q_{min}(G)$. This is a contradiction because $G^{{}^{\prime}}\cong G$. Hence $|x_{g}|>|x_{g-2}|$. And then $a\leq g-1$.

Subcase 2.1 $a\leq g-4$. By Lemma 4.1, without loss of generality, suppose $x_{a+1}\leq x_{a-1}$, $x_{a+1}x_{a}\geq 0$. As Case 1, it is proved that the theorem holds.

Subcase 2.2 $a=g-3$. By Lemma 4.1, suppose $x_{g-2}\leq x_{g-4}$, $x_{g-2}x_{g-3}\geq 0$; suppose $|x_{g-4}|\geq|x_{g-1}|$. Denote by $v_{\tau_{g-3}}$ the pendant vertex attached to $v_{g-3}$. Let $G_{1}=G-v_{g-3}v_{g-4}+v_{g-3}v_{g-1}-v_{g-3}v_{\tau_{g-3}}+v_{g}v_{\tau_{g-3}}$ (if $x_{g-4}\leq x_{g-1}$, let $G_{1}=G-v_{g-2}v_{g-1}+x_{g-2}x_{g-4}$). As Case 1, it is proved that the theorem holds.

Subcase 2.3 $a=g-2$. By Lemma 4.1, suppose $x_{g-1}\leq x_{g-3}$, $x_{g-1}x_{g-2}\geq 0$; suppose $|x_{g-3}|\geq|x_{g}|$. Denote by $v_{\tau_{g-3}}$ the pendant vertex attached to $v_{g-3}$. Let $G_{1}=G-v_{g-2}v_{g-3}+v_{g-2}v_{g}$ (if $x_{g-3}\leq x_{g}$, let $G_{1}=G-v_{g-1}v_{g}+x_{g-1}x_{g-3}-v_{g-3}v_{\tau_{g-3}}+v_{g}v_{\tau_{g-3}}$). As Case 1, it is proved that the theorem holds.

Subcase 2.4 $a=g-1$. Note $|x_{g}|>|x_{g-2}|$. By Lemma 4.1, $x_{g-2}x_{g-1}\geq 0$. Without loss of generality, suppose $x_{g-3}\geq x_{g}$, let $G_{1}=G-v_{g-2}v_{g-3}+v_{g-2}v_{g}$ (if $x_{g-3}\leq x_{g}$, let $G_{1}=G-v_{g-1}v_{g}+x_{g-1}x_{g-3}-v_{g}v_{g+1}+v_{g-3}v_{g+1}$). As Case 1, it is proved that the theorem holds. This completes the proof.      $\Box$

By Lemmas 2.12, 4.7, we get the following Theorem 4.8.

Theorem 4.8

Let $G$ be a nonbipartite connected unicyclic graph of order $n\geq 3$ and with domination number $\frac{n-1}{2}$. Then $q_{min}(G)\geq q_{min}(\mathscr{H}_{3,\frac{n-3}{2}})$ with equality if and only if $G\cong\mathscr{H}_{3,\frac{n-3}{2}}$.

5 Proof of main results

**Proof of Theorem 1.1. ** By Lemmas 2.1, 2.7, then $G$ contains a nonbipartite unicyclic spanning subgraph $H$ with $g_{o}(H)=g_{o}(G)$, $\gamma(H)=\gamma(G)$ and $q_{min}(H)\leq q_{min}(G)$. By Theorem 4.8, it follows that $q_{min}(H)\geq q_{min}(\mathscr{H}_{3,\frac{n-3}{2}})$ with equality if and only if $H\cong\mathscr{H}_{3,\frac{n-3}{2}}$. Thus it follows that $q_{min}(G)\geq q_{min}(\mathscr{H}_{3,\frac{n-3}{2}})$.

Suppose that $q_{min}(G)=q_{min}(\mathscr{H}_{3,\frac{n-3}{2}})$. Then $q_{min}(H)=q_{min}(\mathscr{H}_{3,\frac{n-3}{2}})$ and $H\cong\mathscr{H}_{3,\frac{n-3}{2}}$. For convenience, we suppose that $H=\mathscr{H}_{3,\frac{n-3}{2}}$. Suppose that $Y$ is a unit eigenvector corresponding to $q_{min}(G)$. Note that $q_{min}(\mathscr{H}_{3,\frac{n-3}{2}})=q_{min}(H)\leq Y^{T}Q(H)Y\leq Y^{T}Q(G)Y=q_{min}(G)$. Because we suppose that $q_{min}(G)=q_{min}(\mathscr{H}_{3,\frac{n-3}{2}})$, it follows that $Y^{T}Q(H)Y=Y^{T}Q(G)Y$ and $Q(H)Y=q_{min}(H)Y$.

For $\mathscr{H}_{3,\frac{n-3}{2}}$ (see Fig. 4.2), we claim that $y_{3}>y_{1}$, $y_{3}>y_{2}$. Otherwise, suppose that $y_{3}\leq y_{1}$. Let $H^{\prime}=\mathscr{H}_{3,\frac{n-3}{2}}-v_{3}v_{4}+v_{1}v_{4}$. By Lemma 2.4, it follows that $q_{min}(H^{\prime})<q_{min}(\mathscr{H}_{3,\frac{n-3}{2}})$. This is a contradiction because $H^{\prime}\cong H\cong\mathscr{H}_{3,\frac{n-3}{2}}$. Thus our claim holds.

If $G\neq H$, combined with Lemma 2.3, then for any edge $v_{i}v_{j}\not\in E(H)$, it follows that $x_{i}+x_{j}\neq 0$, and then $Y^{T}Q(H)Y<Y^{T}Q(G)Y$, which contradicts $Y^{T}Q(H)Y=Y^{T}Q(G)Y$. Then it follows that $q_{min}(G)=q_{min}(\mathscr{H}_{3,\frac{n-1}{2}})$ if and only if $G\cong\mathscr{H}_{3,\frac{n-1}{2}}$. This completes the proof.      $\Box$

In a same way, with Lemmas 2.13, 2.14 and 4.6, Theorem 1.2 is proved.

Remark It can be seen that the conjecture in [18] that $\mathbb{S}$ has the smallest $q_{min}$ holds for the graphs with domination number $\gamma=\frac{n-1}{2}$ and the graphs with girth at most 5. With references [17] and [18], it can also be seen that the minimum $q_{min}$ of the connected nonbipartite graph on $n\geq 5$ vertices, with domination number $\frac{n+1}{3}<\gamma\leq\frac{n-2}{2}$ and girth $g\geq 5$, is still open.