The minimum Q-index of strongly connected bipartite digraphs with complete bipartite subdigraphs

TL;DR

This paper investigates the minimum signless Laplacian spectral radius (Q-index) among strongly connected bipartite digraphs containing a complete bipartite subdigraph, identifying the extremal graph with the smallest Q-index.

Contribution

It determines the extremal strongly connected bipartite digraph with the minimum Q-index within a specified class containing a complete bipartite subdigraph.

Findings

Identifies the extremal digraph with the minimum Q-index.

Provides bounds and characterization for the Q-index in this class.

Enhances understanding of spectral properties of bipartite digraphs.

Abstract

Let $\mathcal{G}_{n,p,q}$ denote the set of strongly connected bipartite digraphs on $n$ vertices which contain a complete bipartite subdigraph $\overleftrightarrow{K_{p,q}}$, where $p, q, n$ are positive integers and $p+q \leq n$. In this paper, we study the Q-index (i.e. the signless Laplacian spectral radius) of $\mathcal{G}_{n,p,q}$, and determine the extremal digraph that has the minimum Q-index.

Equations

$$q(B_{n,p,q}^1)\le q(B_{n,p,q}^2),$$

$$(Q(B_{n,p,q}^1)X{)}_i=q(B_{n,p,q}^1)x_i=q{x}_i+\sum _{u\in N_i^{+}}{x}_u=q{x}_i+\sum _{u\in V_q}{x}_u,(i=2,3,\dots ,p),$$

$$x_2=x_3=\dots =x_p\triangleq x_p.$$

$$(Q(B_{n,p,q}^1)X{)}_j=q(B_{n,p,q}^1)x_j=p{x}_j+\sum _{u\in N_j^{+}}{x}_u=p{x}_j+\sum _{u\in V_p}{x}_u,(j=p+1,p+2,\dots ,p+q),$$

$$x_{p+1}=x_{p+2}=\dots =x_{p+q}\triangleq x_{p+1}.$$

$$q(B_{n,p,q}^1)x_1=(q+1)x_1+x_{p+q+1}+q{x}_{p+1},$$

$$q(B_{n,p,q}^1)x_p=q{x}_p+q{x}_{p+1},$$

$$q(B_{n,p,q}^1)x_{p+1}=p{x}_{p+1}+x_1+(p-1)x_p,$$

$$(q(B_{n,p,q}^1)-1{)}^{n-p-q}{x}_{p+q+1}=x_p.$$

$$(q(B_{n,p,q}^1)-q)x_p=q{x}_{p+1}.$$

$$(q(B_{n,p,q}^1)-q)(q(B_{n,p,q}^1)-p)x_p$$

$$(q(B_{n,p,q}^1)-q)(q(B_{n,p,q}^1)-p)(q(B_{n,p,q}^1)-q-1)x_p$$

$$=q(q(B_{n,p,q}^1)-q-1)x_1+q(p-1)(q(B_{n,p,q}^1)-q-1)x_p$$

$$=q{x}_{p+q+1}+q^2{x}_{p+1}+q(p-1)(q(B_{n,p,q}^1)-q-1)x_p(\text{by (refeq:3)})$$

$$=q{x}_{p+q+1}+q(q(B_{n,p,q}^1)-q)x_p+q(p-1)(q(B_{n,p,q}^1)-q-1)x_p(\text{by (refeq:7)})$$

$$=q\frac{1}{(q(B_{n,p,q}^1)-1{)}^{n-p-q}}{x}_p+q(q(B_{n,p,q}^1)-q)x_p+q(p-1)(q(B_{n,p,q}^1)-q-1)x_p,(\text{by (refeq:8)})$$

$$(q(B_{n,p,q}^1)-1{)}^{n-p-q}(q(B_{n,p,q}^1)-q)(q(B_{n,p,q}^1)-p)(q(B_{n,p,q}^1)-q-1)x_p$$

$$=[q+q(q(B_{n,p,q}^1)-q)(q(B_{n,p,q}^1)-1{)}^{n-p-q}$$

$$+q(p-1)(q(B_{n,p,q}^1)-q-1)(q(B_{n,p,q}^1)-1{)}^{n-p-q}]x_p=0.$$

$$(q(B_{n,p,q}^1)-1{)}^{n-p-q}[q^3(B_{n,p,q}^1)-(p+2q+1)q^2(B_{n,p,q}^1)$$

$$+(q^2+pq+p+q)q(B_{n,p,q}^1)-q]-q=0$$

$$(Q(B_{n,p,q}^1)X{)}_1=q(B_{n,p,q}^1)x_1=(q+1)x_1+\sum _{u\in N_1^{+}}{x}_u=(q+1)x_1+x_{p+q+1}+q{x}_{p+1},$$

$$(Q(B_{n,p,q}^1)X{)}_p=q(B_{n,p,q}^1)x_p=q{x}_p+\sum _{u\in N_p^{+}}{x}_u=q{x}_p+q{x}_{p+1},$$

$$(q(B_{n,p,q}^1)-q)(x_1-x_p)=x_1+x_{p+q+1}>0.$$

$$(Q(B_{n,p,q}^1)X{)}_{p+1}=q(B_{n,p,q}^1)x_{p+1}=p{x}_{p+1}+\sum _{u\in N_{p+1}^{+}}{x}_u=p{x}_{p+1}+x_1+(p-1)x_p,$$

$$q(B_{n,p,q}^1)(x_{p+1}-x_p)=p{x}_{p+1}-q{x}_{p+1}+x_1+(p-1-q)x_p.$$

$$q(B_{n,p,q}^1)(x_{p+1}-x_p)>(p-q)(x_{p+1}+x_p)\ge 0.$$

$$q(B_{n,p,q}^3)>q(B_{n,p,q}^1).$$

$$q(B_{n,p,q}^4)>q(B_{n,p,q}^2).$$

$$(Q(B_{n,p,q}^2)X{)}_{p+1}=q(B_{n,p,q}^2)x_{p+1}=(p+1)x_{p+1}+x_{p+q+1}+\sum _{u\in V_p}{x}_u,$$

$$(Q(B_{n,p,q}^2)X{)}_{p+q}=q(B_{n,p,q}^2)x_{p+q}=p{x}_{p+q}+\sum _{u\in V_p}{x}_u,$$

$$(q(B_{n,p,q}^2)-p)(x_{p+1}-x_{p+q})=x_{p+1}+x_{p+q+1}>0$$

$$q(B_{n,p,q}^5)\le q(B_{n,p,q}^6),$$

$$q(B_{n,p,q}^1)<q(B_{n-1,p,q}^5).$$

Full text

The minimum Q-index of strongly connected bipartite digraphs

with complete bipartite subdigraphs ††thanks: Supported by the National Natural Science Foundation of China (No. 11171273) and the Seed Foundation of Innovation and Creation for Graduate Students in Northwestern Polytechnical University (No. Z2017190).

Weige Xi and Ligong Wang111Corresponding author.

Department of Applied Mathematics, School of Science,

Northwestern Polytechnical University, Xi’an, Shaanxi 710072, P.R.China

E-mail: xiyanxwg@163.com, lgwangmath@163.com

Abstract

Let $\mathcal{G}_{n,p,q}$ denote the set of strongly connected bipartite digraphs on $n$ vertices which contain a complete bipartite subdigraph $\overleftrightarrow{K_{p,q}}$, where $p,q,n$ are positive integers and $p+q\leq n$. In this paper, we study the Q-index (i.e. the signless Laplacian spectral radius) of $\mathcal{G}_{n,p,q}$, and determine the extremal digraph that has the minimum Q-index.

Key Words:  Strongly connected bipartite digraphs, Q-index, eigenvector. AMS Subject Classification (2000):  05C50, 15A18.

1 Introduction

Let $G=(V(G),E(G))$ be a digraph with vertex set $V(G)=\{v_{1},v_{2},\ldots,v_{n}\}$ and arc set $E(G)$. If there is an arc from $v_{i}$ to $v_{j}$, we indicate this by writing $(v_{i},v_{j})$, call $v_{j}$ the head of $(v_{i},v_{j})$, and $v_{i}$ the tail of $(v_{i},v_{j})$, respectively. Let $H=(V(H),E(H))$ be a subdigraph of $G$, if $V(H)\subseteq V(G)$, $E(H)\subseteq E(G)$. For any vertex $v_{i}\in V(G)$, $N_{i}^{+}=N_{v_{i}}^{+}(G)=\{v_{j}:(v_{i},v_{j})\in E(G)\}$ is called the set of out-neighbors of $v_{i}$. Let $d_{i}^{+}=|N_{i}^{+}|$ denote the outdegree of the vertex $v_{i}$ in the digraph $G$. A digraph is simple if it has no loops and multiarcs. A digraph is strongly connected if for every pair of vertices $v_{i},v_{j}\in V(G)$, there exists a directed path from $v_{i}$ to $v_{j}$. Let $\overrightarrow{P_{n}}$ and $\overrightarrow{C_{n}}$ denote the directed path and the directed cycle on $n$ vertices, respectively. Suppose $\overrightarrow{P_{k}}=v_{1}v_{2}\dots v_{k}$, we call $v_{1}$ the initial vertex of the directed path $\overrightarrow{P_{k}}$, $v_{k}$ the terminal vertex of the directed path $\overrightarrow{P_{k}}$, respectively.

Let $G=(V(G),E(G))$ be a digraph, If $V(G)=U\cup W$, $U\cap W=\emptyset$ and for any arc $(v_{i},v_{j})\in E(G)$, $v_{i}\in U$ and $v_{j}\in W$ or $v_{i}\in W$ and $v_{j}\in U$, then the digraph $G=(V(G),E(G))$ is called a bipartite digraph. Let $\overleftrightarrow{K_{p,q}}$ be a complete bipartite digraph obtained from a complete bipartite graph $K_{p,q}$ by replacing each edge with a pair of oppositely directed arcs. In this paper, we consider finite, simple strongly connected bipartite digraphs.

For a digraph $G$ of order $n$, let $A(G)=(a_{ij})$ denote the adjacency matrix of $G$, where $a_{ij}=1$ if $(v_{i},v_{j})\in E(G)$ and $a_{ij}=0$ otherwise. Let $D(G)=\textrm{diag}(d_{1}^{+},d_{2}^{+},$ $\ldots,d_{n}^{+})$ be the diagonal matrix with outdegrees of the vertices of $G$ and $Q(G)=D(G)+A(G)$ the signless Laplacian matrix of $G$. The spectral radius of $Q(G)$, i.e., the largest modulus of the eigenvalues of $Q(G)$, is called the Q-index or the signless Laplacian spectral radius of $G$, denoted by $q(G)$. It follows from Perron Frobenius Theorem that $q(G)$ is an eigenvalue of $Q(G)$, and there is a positive unit eigenvector corresponding to $q(G)$ when $G$ is a strongly connected digraph, we called the positive unit eigenvector corresponding to $q(G)$ the Perron vector of digraph $G$.

The problem of determining graphs that maximize or minimize the spectral radius or the largest eigenvalue of the related matrix among a class of given graphs is an important classic problem in spectral graph theory. The problem determining undirected graphs that maximize or minimize of adjacent spectral radius, Laplacian spectral radius and signless Laplacian spectral radius have been well treated in the literature, see [2, 6, 7, 8] and so on, but there is no much known about digraphs. In [5], Lin et al. characterized the extremal digraphs with minimum spectral radius among all digraphs with given clique number and grith, and the extremal digraphs with maximum spectral radius among all digraphs with given vertex connectivity. In [4], Hong and You established some sharp upper or lower bound on the signless Laplacian spectral radius of digraphs with some given parameter such as clique number, girth or vertex connectivity, and characterized the extremal digraph. In [3], Drury and Lin determined the digraphs that have the minimum and second minimum spectral radius among all strongly connected digraphs with given order and dichromatic number. In [1], Chen et.al studied the spectral radius of strongly connected bipartite digraphs which contain a complete bipartite subdigraph and given characterization of the extremal digraph with the least spectral radius.

In this paper, we study the Q-index (i.e. the signless Laplacian spectral radius) of strongly connected bipartite digraphs which have the complete bipartite subdigraph $\overleftrightarrow{K_{p,q}}$  $(p\geq q\geq 1)$, and determine the extremal digraph that has the minimum Q-index.

2 Preliminaries

In this section, we present some known lemmas which are useful for the proof of the main results. In the rest of this paper, let $X=(x_{1},x_{2},\ldots,x_{n})^{T}$ be the Perron vector corresponding to $q(G)$, where $x_{i}$ corresponds to the vertex $v_{i}$.

Lemma 2.1**.**

([4]) Let $G=(V(G),E(G))$ be a simple digraph on $n$ vertices, $u,v,w$ distinct vertices of $V(G)$ and $(u,v)\in E(G)$. Let $H=G-\{(u,v)\}+\{(u,w)\}$. (Noting that if $(u,w)\in E(G)$, then $H$ has multiple arc $(u,w)$.) If $x_{w}\geq x_{v}$, then $q(H)\geq q(G)$. Furthermore, if $H$ is strongly connected and $x_{w}>x_{v}$, then $q(H)>q(G)$.

Lemma 2.2**.**

([4]) Let $G$ be a digraph and $G_{1},G_{2},\ldots,G_{s}$ be the strongly connected components of $G$. Then $q(G)=\max\{q(G_{1}),q(G_{2}),\ldots,q(G_{s})\}$.

Lemma 2.3**.**

([4]) Let $G$ be a digraph and $H$ be a subdigraph of $G$. Then $q(H)\leq q(G)$. If $G$ is strongly connected, and $H$ is a proper subdigraph of $G$, then $q(H)<q(G)$.

Lemma 2.4**.**

([4]) Let $G\ (\neq\overrightarrow{C_{n}})$ be a strongly connected digraph with vertex set $V(G)=\{v_{1},v_{2},\ldots,v_{n}\}$, $\overrightarrow{P_{k}}=v_{1}v_{2}\dots v_{k}\ (k\geq 3)$ be a directed path of $G$ with $d_{i}^{+}=1\ (i=2,3,\ldots,k-1)$. Then we have $x_{2}<x_{3}<\ldots<x_{k-1}<x_{k}$.

Let $G=(V(G),E(G))$ be a digraph with $(u,v)\in E(G)$ and $w\notin V(G)$, $G^{w}=(V(G^{w}),E(G^{w}))$ with $V(G^{w})=V(G)\cup\{w\},\ E(G^{w})=E(G)-\{(u,v)\}+\{(u,w),(w,v)\}$.

Lemma 2.5**.**

([4]) Let $G\ (\neq\overrightarrow{C_{n}})$ be a strongly connected digraph, $w\notin V(G)$, and $G^{w}$ defined as before. Then $q(G)\geq q(G^{w})$.

3 The Q-index of strongly connected

bipartite digraphs which contain a complete bipartite subdigraph

In this section, we will show that if $n\equiv p+q\ (mod\ 2)$ then $B^{5}_{n,p,q}$ is the unique bipartite digraph with the minimum Q-index among all strongly connected bipartite digraphs on $n$ verties which have the complete bipartite subdigraph $\overleftrightarrow{K_{p,q}}$, otherwise, if $n\not\equiv p+q\ (mod\ 2)$ then $B^{1}_{n,p,q}$ is the unique bipartite digraph with the minimum Q-index among all strongly connected bipartite digraphs on $n$ verties which have the complete bipartite subdigraph $\overleftrightarrow{K_{p,q}}\ (p\geq q\geq 1)$.

Let $\overleftrightarrow{K_{p,q}}$ be a complete bipartite digraph with $V(\overleftrightarrow{K_{p,q}})=V_{p}\ \cup\ V_{q}$, $E(\overleftrightarrow{K_{p,q}})=\{(u,v),$ $(v,u)|u\in V_{p},v\in V_{q}\}$ and $|V_{p}|=p$, $|V_{q}|=q$. Let $\mathcal{G}_{n,p,q}$ denote the set of strongly connected bipartite digraphs on $n$ vertices which contain a complete bipartite subdigraph $\overleftrightarrow{K_{p,q}}$. As we all know, if $p+q=n$, then $\mathcal{G}_{n,p,q}=\{\overleftrightarrow{K_{p,q}}\}$, and $q(\overleftrightarrow{K_{p,q}})=p+q$. Thus we only consider the cases when $p+q\leq n-1$ and $p\geq q\geq 1$. In the rest of this section, we just discuss under this assumption.

Let $B^{1}_{n,p,q}=(V(B^{1}_{n,p,q}),E(B^{1}_{n,p,q}))$ be a digraph obtained by adding a directed path $\overrightarrow{P_{n-p-q+2}}=v_{1}v_{p+q+1}v_{p+q+2}\ldots v_{n}v_{p}$ to a complete bipartite digraph $\overleftrightarrow{K_{p,q}}$ such that $V(\overleftrightarrow{K_{p,q}})\cap V(\overrightarrow{P_{n-p-q+2}})=\{v_{1},v_{p}\}$ as shown in Figure 1(a), where $V(B^{1}_{n,p,q})=\{v_{1},v_{2},\ldots,v_{n}\}$. Clearly, if $n-p-q$ is odd, then $B^{1}_{n,p,q}\in\mathcal{G}_{n,p,q}$.

Let $B^{2}_{n,p,q}=(V(B^{2}_{n,p,q}),E(B^{2}_{n,p,q}))$ be a digraph obtained by adding a directed path $\overrightarrow{P_{n-p-q+2}}=v_{p+1}v_{p+q+1}v_{p+q+2}\ldots v_{n}v_{p+q}$ to a complete bipartite digraph $\overleftrightarrow{K_{p,q}}$ such that $V(\overleftrightarrow{K_{p,q}})\cap V(\overrightarrow{P_{n-p-q+2}})=\{v_{p+1},v_{p+q}\}$ as shown in Figure 1(b), where $V(B^{2}_{n,p,q})=\{v_{1},v_{2},$ $\ldots,v_{n}\}$. Clearly, if $n-p-q$ is odd, then $B^{2}_{n,p,q}\in\mathcal{G}_{n,p,q}$.

Let $B^{5}_{n,p,q}=(V(B^{5}_{n,p,q}),E(B^{5}_{n,p,q}))$ be a digraph obtained by adding a directed path $\overrightarrow{P_{n-p-q+2}}=v_{1}v_{p+q+1}v_{p+q+2}\ldots v_{n}v_{p+1}$ to a complete bipartite digraph $\overleftrightarrow{K_{p,q}}$ such that $V(\overleftrightarrow{K_{p,q}})\cap V(\overrightarrow{P_{n-p-q+2}})=\{v_{1},v_{p+1}\}$ as shown in Figure 2(a), where $V(B^{5}_{n,p,q})=\{v_{1},v_{2},\ldots,$ $v_{n}\}$. Clearly, if $n-p-q$ is even, then $B^{1}_{n,p,q}\in\mathcal{G}_{n,p,q}$.

Let $B^{6}_{n,p,q}=(V(B^{6}_{n,p,q}),E(B^{6}_{n,p,q}))$ be a digraph obtained by adding a directed path $\overrightarrow{P_{n-p-q+2}}=v_{p+1}v_{p+q+1}v_{p+q+2}\ldots v_{n}v_{1}$ to a complete bipartite digraph $\overleftrightarrow{K_{p,q}}$ such that $V(\overleftrightarrow{K_{p,q}})\cap V(\overrightarrow{P_{n-p-q+2}})=\{v_{1},v_{p+1}\}$ as shown in Figure 2(b), where $V(B^{6}_{n,p,q})=\{v_{1},v_{2},\ldots,$ $v_{n}\}$. Clearly, if $n-p-q$ is even, then $B^{6}_{n,p,q}\in\mathcal{G}_{n,p,q}$.

Theorem 3.1**.**

For digraphs $B^{1}_{n,p,q}$ and $B^{2}_{n,p,q}$, as shown in Figure 1,

[TABLE]

with equality if and only if $p=q$.

Proof.

If $p=q$, then $B^{2}_{n,p,q}\cong B^{1}_{n,p,q}$. Hence $q(B^{1}_{n,p,q})=(B^{2}_{n,p,q})$. Otherwise $p>q$, let $X=(x_{1},x_{2},\ldots,x_{n})^{T}$ be the Perron vector corresponding to $q(B^{1}_{n,p,q})$ where $x_{i}$ corresponds to the vertex $v_{i}$. Since

[TABLE]

then $(q(B^{1}_{n,p,q})-q)x_{i}=\sum\limits_{u\in V_{q}}x_{u}$. Noting that $\overleftrightarrow{K_{p,q}}$ is a proper subdigraph of $B^{1}_{n,p,q}$ and $B^{1}_{n,p,q}$ is strongly connected, then by Lemma 2.3, we have $q(B^{1}_{n,p,q})>q(\overleftrightarrow{K_{p,q}})=p+q$. Thus

[TABLE]

Since

[TABLE]

similarly, then we have

[TABLE]

From $Q(B^{1}_{n,p,q})X=q(B^{1}_{n,p,q})X$, (1) and (2), we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From (4), we have

[TABLE]

multiply both sides of the equation $(\ref{eq:7})$ by $(q(B^{1}_{n,p,q})-p)$, we have

[TABLE]

multiply both sides of the equation $(\ref{eq:8})$ by $(q(B^{1}_{n,p,q})-q-1)$, we have

[TABLE]

multiply both sides of the above equation by $(q(B^{1}_{n,p,q})-1)^{n-p-q}$, we have

[TABLE]

Thus

[TABLE]

Let $f(x)=(x-1)^{n-p-q}[x^{3}-(p+2q+1)x^{2}+(q^{2}+pq+p+q)x-q]-q$, it is not difficult to see that $q(B^{1}_{n,p,q})$ is the largest real root of $f(x)=0$. Similarly, let $g(x)=(x-1)^{n-p-q}[x^{3}-(q+2p+1)x^{2}+(p^{2}+pq+q+p)x-p]-p$, then $q(B^{2}_{n,p,q})$ is the largest real root of $g(x)=0$. Since $p>q$, thus $f(x)-g(x)=(x-1)^{n-p-q}(p-q)[x^{2}-(p+q)x+1]+p-q>0$, for all $x>p+q$. Since $q(B^{1}_{n,p,q})>p+q$ and $q(B^{2}_{n,p,q})>p+q$, then we have $q(B^{1}_{n,p,q})<q(B^{2}_{n,p,q})$. Therefore, $q(B^{1}_{n,p,q})\leq q(B^{2}_{n,p,q})$ with equality if and only if $p=q$. ∎

Lemma 3.2**.**

Let $X=(x_{1},x_{2},\ldots,x_{n})^{T}$ be the Perron vector corresponding to $q(B^{1}_{n,p,q})$, where $x_{i}$ corresponds to the vertex $v_{i}$, then we have

(i) $x_{1}>x_{p}$;

(ii) $x_{p+1}>x_{p}$.

Proof.

(i) By Theorem 3.1, we have

[TABLE]

[TABLE]

From (9) and (10), then we have

[TABLE]

Thus $x_{1}>x_{p}$ by $q(B^{1}_{n,p,q})>p+q$.

(ii) Since

[TABLE]

From (10) and (11), then we have

[TABLE]

Since $x_{1}>x_{p}$, then

[TABLE]

Thus $x_{p+1}>x_{p}$ by $q(B^{1}_{n,p,q})>p+q$. ∎

Theorem 3.3**.**

Let $B^{3}_{n,p,q}=B^{1}_{n,p,q}-\{(v_{n},v_{p})\}+\{(v_{n},v_{1})\}$. Then

[TABLE]

Proof.

Clearly $B^{3}_{n,p,q}$ is strongly connected. Let $X=(x_{1},x_{2},\ldots,x_{n})^{T}$ be the Perron vector corresponding to $q(B^{1}_{n,p,q})$, where $x_{i}$ corresponds to the vertex $v_{i}$. Thus $q(B^{3}_{n,p,q})>q(B^{1}_{n,p,q})$ by Lemmas 2.1 and 3.2. ∎

Theorem 3.4**.**

Let $B^{4}_{n,p,q}=B^{2}_{n,p,q}-\{(v_{n},v_{p+q})\}+\{(v_{n},v_{p+1})\}$. Then

[TABLE]

Proof.

Clearly $B^{4}_{n,p,q}$ is strongly connected. Let $X=(x_{1},x_{2},\ldots,x_{n})^{T}$ be the Perron vector corresponding to $q(B^{2}_{n,p,q})$, where $x_{i}$ corresponds to the vertex $v_{i}$. By Lemma 2.1, we only need to prove $x_{p+1}>x_{p+q}$.

[TABLE]

[TABLE]

From (12) and (13), then we have

[TABLE]

Thus $x_{p+1}>x_{p+q}$ by $q(B^{2}_{n,p,q})>p+q$. ∎

As the proof of the following result is similar to that of Theorem 3.1, we omit the details.

Theorem 3.5**.**

For digraphs $B^{5}_{n,p,q}$ and $B^{6}_{n,p,q}$, as shown in Figure 2,

[TABLE]

with equality if and only if $p=q$.

Theorem 3.6**.**

For digraphs $B^{1}_{n,p,q}$ and $B^{5}_{n,p,q}$, as shown in Figure 1 and 2,

[TABLE]

Proof.

Since $B^{5}_{n,p,q}=B^{1}_{n,p,q}-\{(v_{n},v_{p})\}+\{(v_{n},v_{p+1})\}$ and $B^{5}_{n,p,q}$ is strongly connected. Let $X=(x_{1},x_{2},\ldots,x_{n})^{T}$ be the Perron vector corresponding to $q(B^{1}_{n,p,q})$, where $x_{i}$ corresponds to the vertex $v_{i}$. By Lemma 3.2, we have $x_{p+1}>x_{p}$. Thus by Lemma 2.1, $q(B^{5}_{n,p,q})>q(B^{1}_{n,p,q})$. Then $q(B^{5}_{n-1,p,q})>q(B^{5}_{n,p,q})$ by Lemma 2.5. Therefore, we have $q(B^{1}_{n,p,q})<q(B^{5}_{n-1,p,q})$. ∎

Theorem 3.7**.**

For digraphs $B^{1}_{n,p,q}$ and $B^{5}_{n,p,q}$, as shown in Figure 1 and 2,

[TABLE]

Proof.

Let $B^{5*}_{n,p,q}=B^{5}_{n,p,q}-\{(v_{n-1},v_{n})\}+\{(v_{n-1},v_{p})\}$ and $X=(x_{1},x_{2},\ldots,x_{n})^{T}$ be the Perron vector corresponding to $q(B^{5}_{n,p,q})$, where $x_{i}$ corresponds to the vertex $v_{i}$. Since

[TABLE]

[TABLE]

Then from $Q(B^{5}_{n,p,q})X=q(B^{5}_{n,p,q})X$, we have

[TABLE]

[TABLE]

thus $(q(B^{5}_{n,p,q})-1)qx_{n}=qx_{p+1}=(q(B^{5}_{n,p,q})-q)x_{p}$. Since $q(B^{5}_{n,p,q})-1\geq q(B^{5}_{n,p,q})-q>0$ by $q(B^{5}_{n,p,q})>p+q$, then $x_{n}\leq x_{p}$ By Lemma 2.1, we have $q(B^{5*}_{n,p,q})\geq q(B^{5}_{n,p,q})$. Since $B^{1}_{n-1,p,q}$ and $K_{1}=\{v_{n}\}$ are the two strongly connected components of $B^{5*}_{n,p,q}$, then by Lemma 2.2, we have $q(B^{5*}_{n,p,q})=\max\{q(B^{1}_{n-1,p,q}),q(K_{1})\}=q(B^{1}_{n-1,p,q})$. Thus $q(B^{5}_{n,p,q})\leq q(B^{1}_{n-1,p,q})$. ∎

Theorem 3.8**.**

Let $p\geq q\geq 1$, $p+q\leq n-1$, $n\equiv p+q\ (mod\ 2)$ and $G\in\mathcal{G}_{n,p,q}$ be a bipartite digraph, then $q(G)\geq q(B^{5}_{n,p,q})$ and the equality holds if and only if $G\cong B^{5}_{n,p,q}$.

Proof.

Clearly, $\overleftrightarrow{K_{p,q}}$ is a proper subdigraph of $G$ since $G\in\mathcal{G}_{n,p,q}$. Since $G$ is strongly connected, it is possible to obtain a digraph $H$ from $G$ by deleting vertices and arcs in a way such that one has a subdigraph $\overleftrightarrow{K_{p,q}}$. Therefore

(1) $H\cong B^{1}_{p+q+k,p,q}$,     ($k\equiv 1$  (mod  2), $k\geq 1$) or

(2) $H\cong B^{2}_{p+q+k,p,q}$,     ($k\equiv 1$  (mod  2), $k\geq 1$) or

(3) $H\cong B^{3}_{p+q+k,p,q}$,     ($k\equiv 1$  (mod  2), $k\geq 1$) or

(4) $H\cong B^{4}_{p+q+k,p,q}$,     ($k\equiv 1$  (mod  2), $k\geq 1$) or

(5) $H\cong B^{5}_{p+q+l,p,q}$,     ($l\equiv 0$  (mod  2), $l\geq 2$) or

(6) $H\cong B^{6}_{p+q+l,p,q}$,     ($l\equiv 0$  (mod  2), $l\geq 2$).

By Lemma 2.3, $q(H)\leq q(G)$, the equality holds if and only if $H\cong G$.

Case (i). $H\cong B^{1}_{p+q+k,p,q}$,     ($k\equiv 1$  (mod  2), $k\geq 1$)

Insert $n-p-q-k-1$ vertices into the directed $\overrightarrow{P_{k+2}}$ such that the resulting bipartite digraphs is $B^{1}_{n-1,p,q}$, then $q(B^{1}_{n-1,p,q})\leq q(H)$ by using Lemma 2.5 repeatedly $n-p-q-k-1$ times, and thus $q(B^{5}_{n,p,q})\leq q(B^{1}_{n-1,p,q})\leq q(H)<q(G)$ by Theorem 3.7.

Case (ii). $H\cong B^{2}_{p+q+k,p,q}$,     ($k\equiv 1$  (mod  2), $k\geq 1$)

Insert $n-p-q-k-1$ vertices into the directed $\overrightarrow{P_{k+2}}$ such that the resulting bipartite digraphs is $B^{2}_{n-1,p,q}$, then $q(B^{2}_{n-1,p,q})\leq q(H)$ by using Lemma 2.5 repeatedly $n-p-q-k-1$ times, and thus $q(B^{5}_{n,p,q})\leq q(B^{1}_{n-1,p,q})\leq q(B^{2}_{n-1,p,q})\leq q(H)<q(G)$ by Theorems 3.1 and 3.7.

Case (iii). $H\cong B^{3}_{p+q+k,p,q}$,     ($k\equiv 1$  (mod  2), $k\geq 1$)

Insert $n-p-q-k-1$ vertices into the directed $\overrightarrow{C_{k+1}}$ such that the resulting bipartite digraphs is $B^{3}_{n-1,p,q}$, then $q(B^{3}_{n-1,p,q})\leq q(H)$ by using Lemma 2.5 repeatedly $n-p-q-k-1$ times, and thus $q(B^{5}_{n,p,q})\leq q(B^{1}_{n-1,p,q})<q(B^{3}_{n-1,p,q})\leq q(H)<q(G)$ by Theorems 3.3 and 3.7.

Case (iv). $H\cong B^{4}_{p+q+k,p,q}$,     ($k\equiv 1$  (mod  2), $k\geq 1$)

Insert $n-p-q-k-1$ vertices into the directed $\overrightarrow{C_{k+1}}$ such that the resulting bipartite digraphs is $B^{4}_{n-1,p,q}$, then $q(B^{4}_{n-1,p,q})\leq q(H)$ by using Lemma 2.5 repeatedly $n-p-q-k-1$ times, and thus $q(B^{5}_{n,p,q})\leq q(B^{1}_{n-1,p,q})\leq q(B^{2}_{n-1,p,q})\leq q(B^{4}_{n-1,p,q})\leq q(H)<q(G)$ by Theorems 3.1, 3.4 and 3.7.

Case (v). $H\cong B^{5}_{p+q+l,p,q}$,     ($l\equiv 0$  (mod  2), $l\geq 2$)

Insert $n-p-q-l$ vertices into the directed $\overrightarrow{P_{l+2}}$ such that the resulting bipartite digraphs is $B^{5}_{n,p,q}$, then $q(B^{5}_{n,p,q})\leq q(H)$ by using Lemma 2.5 repeatedly $n-p-q-l$ times, and thus $q(B^{5}_{n,p,q})\leq q(H)\leq q(G)$.

Case (vi). $H\cong B^{6}_{p+q+l,p,q}$,     ($l\equiv 0$  (mod  2), $l\geq 2$)

Insert $n-p-q-l$ vertices into the directed $\overrightarrow{P_{l+2}}$ such that the resulting bipartite digraphs is $B^{6}_{n,p,q}$, then $q(B^{6}_{n,p,q})\leq q(H)$ by using Lemma 2.5 repeatedly $n-p-q-l$ times, and thus $q(B^{5}_{n,p,q})\leq q(B^{6}_{n,p,q})\leq q(H)\leq q(G)$ by Theorem 3.5.

Combining the above six cases, we have $q(G)\geq q(B^{5}_{n,p,q})$ and the equality holds if and only if $G\cong B^{5}_{n,p,q}$, where $n\equiv p+q\ (mod\ 2)$. ∎

Theorem 3.9**.**

Let $p\geq q\geq 1$, $p+q\leq n-1$, $n\not\equiv p+q\ (mod\ 2)$ and $G\in\mathcal{G}_{n,p,q}$ be a bipartite digraph, then $q(G)\geq q(B^{1}_{n,p,q})$ and the equality holds if and only if $G\cong B^{1}_{n,p,q}$.

Proof.

Clearly, $\overleftrightarrow{K_{p,q}}$ is a proper subdigraph of $G$ since $G\in\mathcal{G}_{n,p,q}$. Since $G$ is strongly connected, it is possible to obtain a digraph $H$ from $G$ by deleting vertices and arcs in a way such that one has a subdigraph $\overleftrightarrow{K_{p,q}}$. Therefore

(1) $H\cong B^{1}_{p+q+k,p,q}$,     ($k\equiv 1$  (mod  2), $k\geq 1$) or

(2) $H\cong B^{2}_{p+q+k,p,q}$,     ($k\equiv 1$  (mod  2), $k\geq 1$) or

(3) $H\cong B^{3}_{p+q+k,p,q}$,     ($k\equiv 1$  (mod  2), $k\geq 1$) or

(4) $H\cong B^{4}_{p+q+k,p,q}$,     ($k\equiv 1$  (mod  2), $k\geq 1$) or

(5) $H\cong B^{5}_{p+q+l,p,q}$,     ($l\equiv 0$  (mod  2), $l\geq 2$) or

(6) $H\cong B^{6}_{p+q+l,p,q}$,     ($l\equiv 0$  (mod  2), $l\geq 2$).

By Lemma 2.3, $q(H)\leq q(G)$, the equality holds if and only if $H\cong G$.

Case (i). $H\cong B^{1}_{p+q+k,p,q}$,     ($k\equiv 1$  (mod  2), $k\geq 1$)

Insert $n-p-q-k$ vertices into the directed $\overrightarrow{P_{k+2}}$ such that the resulting bipartite digraphs is $B^{1}_{n,p,q}$, then $q(B^{1}_{n,p,q})\leq q(H)$ by using Lemma 2.5 repeatedly $n-p-q-k$ times, and thus $q(B^{1}_{n,p,q})\leq q(H)\leq q(G)$.

Case (ii). $H\cong B^{2}_{p+q+k,p,q}$,     ($k\equiv 1$  (mod  2), $k\geq 1$)

Insert $n-p-q-k$ vertices into the directed $\overrightarrow{P_{k+2}}$ such that the resulting bipartite digraphs is $B^{2}_{n,p,q}$, then $q(B^{2}_{n,p,q})\leq q(H)$ by using Lemma 2.5 repeatedly $n-p-q-k$ times, and thus $q(B^{1}_{n,p,q})\leq q(B^{2}_{n,p,q})\leq q(H)\leq q(G)$ by Theorem 3.1.

Case (iii). $H\cong B^{3}_{p+q+k,p,q}$,     ($k\equiv 1$  (mod  2), $k\geq 1$)

Insert $n-p-q-k$ vertices into the directed $\overrightarrow{C_{k+1}}$ such that the resulting bipartite digraphs is $B^{3}_{n,p,q}$, then $q(B^{3}_{n,p,q})\leq q(H)$ by using Lemma 2.5 repeatedly $n-p-q-k$ times, and thus $q(B^{1}_{n,p,q})<q(B^{3}_{n,p,q})\leq q(H)\leq q(G)$ by Theorem 3.3.

Case (iv). $H\cong B^{4}_{p+q+k,p,q}$,     ($k\equiv 1$  (mod  2), $k\geq 1$)

Insert $n-p-q-k$ vertices into the directed $\overrightarrow{C_{k+1}}$ such that the resulting bipartite digraphs is $B^{4}_{n-1,p,q}$, then $q(B^{4}_{n,p,q})\leq q(H)$ by using Lemma 2.5 repeatedly $n-p-q-k$ times, and thus $q(B^{1}_{n,p,q})\leq q(B^{2}_{n,p,q})<q(B^{4}_{n,p,q})\leq q(H)\leq q(G)$ by Theorems 3.1 and 3.4.

Case (v). $H\cong B^{5}_{p+q+l,p,q}$,     ($l\equiv 0$  (mod  2), $l\geq 2$)

Insert $n-p-q-l-1$ vertices into the directed $\overrightarrow{P_{l+2}}$ such that the resulting bipartite digraphs is $B^{5}_{n-1,p,q}$, then $q(B^{5}_{n-1,p,q})\leq q(H)$ by using Lemma 2.5 repeatedly $n-p-q-l-1$ times, and thus $q(B^{1}_{n,p,q})<q(B^{5}_{n-1,p,q})\leq q(H)<q(G)$ by Theorem 3.6.

Case (vi). $H\cong B^{6}_{p+q+l,p,q}$,     ($l\equiv 0$  (mod  2), $l\geq 2$)

Insert $n-p-q-l-1$ vertices into the directed $\overrightarrow{P_{l+2}}$ such that the resulting bipartite digraphs is $B^{6}_{n-1,p,q}$, then $q(B^{6}_{n-1,p,q})\leq q(H)$ by using Lemma 2.5 repeatedly $n-p-q-l-1$ times, and thus $q(B^{1}_{n,p,q})<q(B^{5}_{n-1,p,q})\leq q(B^{6}_{n-1,p,q})\leq q(H)<q(G)$ by Theorems 3.5 and 3.6.

Combining the above six cases, we have $q(G)\geq q(B^{1}_{n,p,q})$ and the equality holds if and only if $G\cong B^{1}_{n,p,q}$, where $n\not\equiv p+q\ (mod\ 2)$. ∎