Results on resistance distance and Kirchhoff index of graphs with generalized pockets

TL;DR

This paper derives explicit formulas for resistance distance and Kirchhoff index in graphs with generalized pockets, linking these measures to the properties of the factor graph, thus advancing graph analysis techniques.

Contribution

It introduces closed-form formulas for resistance distance and Kirchhoff index in graphs with generalized pockets, connecting them to the factor graph's metrics.

Findings

Formulas express resistance distance in terms of the factor graph.

Formulas relate Kirchhoff index to the factor graph.

Provides tools for analyzing complex graph structures.

Abstract

In this paper, we derive closed-form formulas for resistance distance and Kirchhoff index of graphs with generalized pockets in terms of the resistance distance and Kirchhoff index of the factor graph.

Equations

$$r_{uv}(G)=(L_G^{(1)}{)}_{uu}+(L_G^{(1)}{)}_{vv}-(L_G^{(1)}{)}_{uv}-(L_G^{(1)}{)}_{vu}=(L_G^{\#}{)}_{uu}+(L_G^{\#}{)}_{vv}-2(L_G^{\#}{)}_{uv}.$$

$$\begin{array}[]{crl}M=\Bigg{(}\begin{array}[]{cccccccccccccccc}A&B\\ C&D\\ \end{array}\Bigg{)}\end{array}$$

$$\begin{array}[]{crl}M^{-1}&=&\Bigg{(}\begin{array}[]{cccccccccccccccc}A^{-1}+A^{-1}BS^{-1}CA^{-1}&-A^{-1}BS^{-1}\\ -S^{-1}CA^{-1}&S^{-1}\\ \end{array}\Bigg{)}\\ &=&\Bigg{(}\begin{array}[]{cccccccccccccccc}(A-BD^{-1}C)^{-1}&-A^{-1}BS^{-1}\\ -S^{-1}CA^{-1}&S^{-1}\\ \end{array}\Bigg{)},\end{array}$$

$$(L+a{I}_n-\frac{a}{n}{j}_{n\times n}{)}^{\#}=(L+aI{)}^{-1}-\frac{1}{an}{j}_{n\times n}.$$

$$Kf(G)=ntr(L_G^{(1)})-1^T{L}_G^{(1)}1=ntr(L_G^{\#}).$$

$$\begin{array}[]{crl}L=\left(\begin{array}[]{cccccccccccccccc}A&B\\ B^{T}&D\\ \end{array}\right)\end{array}$$

$$\begin{array}[]{crl}X=\left(\begin{array}[]{cccccccccccccccc}H^{\#}&-H^{\#}BD^{-1}\\ -D^{-1}B^{T}H^{\#}&D^{-1}+D^{-1}B^{T}H^{\#}BD^{-1}\\ \end{array}\right)\end{array}$$

$$r_{ij}(G[F;H_v,l])=(L^{\#}(F){)}_{ii}+(L^{\#}(F){)}_{jj}-2(L^{\#}(F){)}_{ij}.$$

$$r_{ij}(G[F;H_v,l])$$

$$=$$

$$r_{ij}(G[F;H_v,l])$$

$$=$$

$$\begin{array}[]{crl}L(G)=\left(\begin{array}[]{ccccccc}L(F)+lI_{n}&-1_{l}^{T}\otimes I_{n}&0\\ -1_{l}\otimes I_{n}&(L(H_{1})+(m-l+1)I_{l})\otimes I_{n}&-J_{l\times(m-l)}\otimes I_{n}\\ 0&-J_{(m-l)\times l}\otimes I_{n}&(L(H_{2})+lI_{m-l})\otimes I_{n}\\ \end{array}\right).\end{array}$$

$$\begin{array}[]{crl}D=\left(\begin{array}[]{cccccccccccccccc}(L(H_{1})+(m-l+1)I_{l})\otimes I_{n}&-J_{l\times(m-l)}\otimes I_{n}\\ -J_{(m-l)\times l}\otimes I_{n}&(L(H_{2})+lI_{m-l})\otimes I_{n}\\ \end{array}\right)\end{array}.$$

$$\begin{array}[]{crl}A_{1}-B_{1}D_{1}^{-1}C_{1}&=&\left(L(H_{1})+(m-l+1)I_{l}\right)\otimes I_{n}-(-J_{l\times(m-l)}\otimes I_{n})((L(H_{2})+lI_{m-l})^{-1}\otimes I_{n})\\ &&(-J_{(m-l)\times l}\otimes I_{n})\\ &=&(L(H_{1})+(m-l+1)I_{l})\otimes I_{n}-1_{l}[1^{T}_{m-l}(L(H_{2})+lI_{m-l})^{-1}1_{m-l}]1^{T}_{l}\otimes I_{n}\\ &=&(L(H_{1})+(m-l+1)I_{l})\otimes I_{n}-\frac{m-l}{l}J_{l\times l}\otimes I_{n}\\ &=&[L(H_{1})+(m-l+1)I_{l}-\frac{m-l}{l}J_{l\times l}]\otimes I_{n},\end{array}$$

$$\begin{array}[]{crl}S^{-1}&=&(D_{1}-C_{1}A_{1}^{-1}B_{1})^{-1}\\ &=&[\left(L(H_{2})+lI_{m-l}\right)\otimes I_{n}-(-J_{(m-l)\times l}\otimes I_{n})((L(H_{1})+(m-l+1)I_{l})^{-1}\otimes I_{n})\\ &&(-J_{l\times(m-l)}\otimes I_{n})]^{-1}\\ &=&[(L(H_{2})+lI_{m-l})\otimes I_{n}-(J_{(m-l)\times l}(L(H_{1})+(m-l+1)I_{l})^{-1}J_{l\times(m-l)})\otimes I_{n}]^{-1}\\ &=&(L(H_{2})+lI_{m-l}-\frac{l}{m-l+1}J_{(m-l)\times(m-l)})^{-1}\otimes I_{n}.\end{array}$$

$$\begin{array}[]{crl}-A_{1}^{-1}B_{1}S^{-1}&=&-[(L(H_{1})+(m-l+1)I_{l})^{-1}\otimes I_{n}](-J_{l\times(m-l)}\otimes I_{n})\\ &&[L(H_{2})+lI_{m-l}-\frac{l}{m-l+1}J_{(m-l)\times(m-l)}]^{-1}\otimes I_{n}\\ &=&\frac{1}{m-l+1}1_{l}\times\frac{m-l+1}{l}1^{T}_{m-l}\otimes I_{n}\\ &=&\frac{1}{l}J_{l\times(m-l)}\otimes I_{n}.\end{array}$$

$$=$$

$$\begin{array}[]{crl}H&=&A-BD^{-1}B^{T}\\ &=&L(F)+lI_{n}-\left(\begin{array}[]{cccccccccccccccc}-1_{l}^{T}\otimes I_{n}&0\\ \end{array}\right)\\ &&\left(\begin{array}[]{cccccccccccccccc}[(L(H_{1})+(m-l+1)I_{l}-\frac{m-l}{l}J_{l\times l}]^{-1}\otimes I_{n}&\frac{1}{l}J_{l\times(m-l)}\otimes I_{n}\\ \frac{1}{l}J_{(m-l)\times l}\otimes I_{n}&[L(H_{2})+lI_{m-l}-\frac{l}{m-l+1}J_{(m-l)\times(m-l)}]^{-1}\otimes I_{n}\\ \end{array}\right)\\ &&\left(\begin{array}[]{cccccccccccccccc}-1_{l}\otimes I_{n}\\ 0\\ \end{array}\right)\\ &=&L(F)+lI_{n}-(1_{l}^{T}\otimes I_{n})[L(H_{1})+(m-l+1)I_{l}-\frac{m-l}{l}J_{l\times l}]^{-1}\otimes I_{n}](1_{l}\otimes I_{n})\\ &=&L(F)+lI_{n}-lI_{n}\\ &=&L(F),\end{array}$$

$$\begin{array}[]{crl}&=&-L^{\#}(F)\left(\begin{array}[]{cccccccccccccccc}-1_{l}^{T}\otimes I_{n}&0\\ \end{array}\right)\\ &&\left(\begin{array}[]{cccccccccccccccc}[(L(H_{1})+(m-l+1)I_{l}-\frac{m-l}{l}J_{l\times l}]^{-1}\otimes I_{n}&\frac{1}{l}J_{l\times(m-l)}\otimes I_{n}\\ \frac{1}{l}J_{(m-l)\times l}\otimes I_{n}&[(L(H_{2})+lI_{m-l}-\frac{l}{m-l+1}J_{(m-l)\times(m-l)}]^{-1}\otimes I_{n}\\ \end{array}\right)\\ &=&\left(L^{\#}(F)(1^{T}_{l}\otimes I_{n}),L^{\#}(F)(1^{T}_{m-l}\otimes I_{n})\right)\end{array}$$

$$\begin{array}[]{crl}-D^{-1}B^{T}H^{\#}&=&\left(\begin{array}[]{cccccccccccccccc}(1_{l}\otimes I_{n})L^{\#}(F)&\\ (1_{m-l}\otimes I_{n})L^{\#}(F)&\\ \end{array}\right).\end{array}$$

$$\begin{array}[]{crl}D^{-1}B^{T}H^{\#}BD^{-1}&=&\left(\begin{array}[]{cccccccccccccccc}(1_{l}\otimes I_{n})L^{\#}(F)&\\ (1_{m-l}\otimes I_{n})L^{\#}(F)&\\ \end{array}\right)\left((1^{T}_{l}\otimes I_{n}),(1^{T}_{m-l}\otimes I_{n})\right)\\ &=&\left(\begin{array}[]{cccccccccccccccc}(1_{l}\otimes I_{n})L^{\#}(F)(1_{l}^{T}\otimes I_{n})&(1_{l}\otimes I_{n})L^{\#}(F)(1_{m-l}^{T}\otimes I_{n})\\ (1_{m-l}\otimes I_{n})L^{\#}(F)(1^{T}_{l}\otimes I_{n})&(1_{m-l}\otimes I_{n})L^{\#}(F)(1^{T}_{m-l}\otimes I_{n})\\ \end{array}\right).\\ \end{array}\\$$

$$\displaystyle\left(\begin{array}[]{cccccccccccccccc}L^{\#}(F)&L^{\#}(F)(1^{T}_{l}\otimes I_{n})&L^{\#}(F)(1^{T}_{m-l}\otimes I_{n})\\ (1_{l}\otimes I_{n})L^{\#}(F)&P^{-1}+(1_{l}\otimes I_{n})L^{\#}(F)(1^{T}_{l}\otimes I_{n})&M\\ (1_{m-l}\otimes I_{n})L^{\#}(F)&M^{T}&Q^{-1}+(1_{m-l}\otimes I_{n})L^{\#}(F)(1^{T}_{m-l}\otimes I_{n})\\ \end{array}\right)$$

$$r_{ij}(G[F;H_v,l])=(L^{\#}(F){)}_{ii}+(L^{\#}(F){)}_{jj}-2(L^{\#}(F){)}_{ij}=r_{ij}(F),$$

$$r_{ij}(G[F;H_v,l])$$

$$=$$

Full text

Keywords: Kirchhoff index, Resistance distance, Generalized inverse

AMS Mathematics Subject Classification(2000): 05C50; O157.5

1 Introduction

All graphs considered in this paper are simple and undirected. The resistance distance between vertices $u$ and $v$ of $G$ was defined by Klein and Randi$\acute{c}$ [1] to be the effective resistance between nodes $u$ and $v$ as computed with Ohm’s law when all the edges of $G$ are considered to be unit resistors. The Kirchhoff index $Kf(G)$ was defined in [1] as $Kf(G)=\sum_{u<v}r_{uv}$, where $r_{uv}(G)$ denote the resistance distance between $u$ and $v$ in $G$. Resistance distance are, in fact, intrinsic to the graph, with some nice purely mathematical interpretations and other interpretations. The Kirchhoff index was introduced in chemistry as a better alternative to other parameters used for discriminating different molecules with similar shapes and structures [1]. The resistance distance and the Kirchhoff index have attracted extensive attention due to its wide applications in physics, chemistry and others. Up till now, many results on the resistance distance and the Kirchhoff index are obtained. See $(\cite[cite]{[\@@bibref{}{JZ}{}{}]},\cite[cite]{[\@@bibref{}{JB1}{}{}]},\cite[cite]{[\@@bibref{}{JB2}{}{}]},\cite[cite]{[\@@bibref{}{SWZhB}{}{}]})$ and the references therein to know more. However, the resistance distance and Kirchhoff index of the graph is, in general, a difficult thing from the computational point of view. Therefore, the bigger is the graph, the more difficult is to compute the resistance distance and Kirchhoff index, so a common strategy is to consider complex graph as composite graph, and to find relations between the resistance distance and Kirchhoff index of the original graphs.

Let $G=(V(G),E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. Let $d_{i}$ be the degree of vertex $i$ in $G$ and $D_{G}=diag(d_{1},d_{2},\cdots d_{|V(G)|})$ the diagonal matrix with all vertex degrees of $G$ as its diagonal entries. For a graph $G$, let $A_{G}$ and $B_{G}$ denote the adjacency matrix and vertex-edge incidence matrix of $G$, respectively. The matrix $L_{G}=D_{G}-A_{G}$ is called the Laplacian matrix of $G$, where $D_{G}$ is the diagonal matrix of vertex degrees of $G$. We use $\mu_{1}(G)\geq u_{2}(G)\geq\cdots\geq\mu_{n}(G)=0$ to denote the eigenvalues of $L_{G}$. For other undefined notations and terminology from graph theory, the readers may refer to [6] and the references therein $(\cite[cite]{[\@@bibref{}{ChZh}{}{}]}-\cite[cite]{[\@@bibref{}{YK}{}{}]})$.

The computation of resistance distance between two nodes in a resistor network is a classical problem in electric theory and graph theory. For certain families of graphs it is possible to identify a graph by looking at the resistance distance and Kirchhoff index. More generally, this is not possible. In some cases, the resistance distance and Kirchhoff index of a relatively larger graph can be described in terms of the resistance distance and Kirchhoff index of some smaller(and simpler) graphs using some simple graph operations. There are results that discuss the resistance distance and Kirchhoff index of graphs obtained by means of some operations on graphs like join, graph products, corona and many variants of corona(like edge corona, neighborhood corona, edge, neighborhood corona,etc.). For such operations often it is possible to describe the resistance distance and Kirchhoff index of the resulting graph using the resistance distance and Kirchhoff index of the corresponding constituting graph, see $(\cite[cite]{[\@@bibref{}{BYZhZH}{}{}]},\cite[cite]{[\@@bibref{}{LZhB}{}{}]})$ for reference. This paper consider the resistance distance and Kirchhoff index of the graph operations below, which come from [11].

Definition 1 [11] Let $F$, $H_{v}$ be connected graphs, $v$ be a specified vertex of $H_{v}$ and $u_{1},u_{2},...,u_{k}\in F$. Let $G=G[F,u_{1},u_{2},...,u_{k},H_{v}]$ be the graph obtained by taking one copy of $F$ and $k$ copies of $H_{v}$, and then attaching the $i$th copy of $H_{v}$ to the vertex $u_{i}$, $i=1,2,...,k$, at the vertex $v$ of $H_{v}$(identify $u_{i}$ with the vertex$v$ of the $i$th copy). Then the copies of the graph $H_{v}$ that are attached to the vertices $u_{i}$, $i=1,2,...,k$ are referred to as pockets, and $G$ is described as a graph with $k$ pockets.

Barik [11] has described the $L$-spectrum of $G=G[F,u_{1},u_{2},...,u_{k},H_{v}]$ using the $L$-spectrum of $F$ and $H_{v}$ in a particular case when $deg(v)=m$. Recently Barik and Sahoo [12] have described the Laplacian spectrum of more such graphs relaxing condition $deg(v)=m$. Let $deg(v)=l$, $1\leq l\leq m$. In this case, we denote $G=G[F,u_{1},u_{2},...,u_{k},H_{v}]$ more precisely by $G=G[F,u_{1},u_{2},...,u_{k};H_{v},l]$. When $k=n$, we denote simply by $G=G[F;H_{v},l]$. If $deg(v)=l$, $1\leq l\leq m$, let $N(v)=\{v_{1},v_{2},\ldots,v_{l}\}\subset V(H_{v})$ be the neighbourhood set of $v$ in $H_{v}$. Let $H_{1}$ be the subgraph of $H(v)$ induced by the vertices in $N(v)$ and $H_{2}$ be the subgraph of $H_{v}$ induced by the vertices which are in $V(H_{v})\backslash(N(v)\cup\{v\})$. When $H_{v}=H_{1}\vee(H_{2}+\{v\})$, we describe the resistance distance and Kirchhoff index of $G=G[F,u_{1},u_{2},...,u_{k},H_{v}]$. The results are contained in Section 3 of the article.

Further, when $F=F_{1}\vee F_{2}$, where $F_{1}$ is the subgraph of $F$ induced by the vertices $u_{1},u_{2},\ldots,u_{k}$ and $F_{2}$ is the subgraph of $F$ induced by the vertices $u_{k+1},u_{k+2},\ldots,u_{n}$. In this case, we describe the resistance distance and Kirchhoff index of $G[F,u_{1},u_{2},...,u_{k};H_{v},l]$. These results are contained in Section 4.

2 Preliminaries

The $\{1\}$-inverse of $M$ is a matrix $X$ such that $MXM=M$. If $M$ is singular, then it has infinite $\{1\}$-inverse [16]. For a square matrix $M$, the group inverse of $M$, denoted by $M^{\#}$, is the unique matrix $X$ such that $MXM=M$, $XMX=X$ and $MX=XM$. It is known that $M^{\#}$ exists if and only if $rank(M)=rank(M^{2})$ $(\cite[cite]{[\@@bibref{}{A. Ben-Israel}{}{}]},\cite[cite]{[\@@bibref{}{BSZHW}{}{}]})$. If $M$ is real symmetric, then $M^{\#}$ exists and $M^{\#}$ is a symmetric $\{1\}$-inverse of $M$. Actually, $M^{\#}$ is equal to the Moore-Penrose inverse of $M$ since $M$ is symmetric [17].

It is known that resistance distances in a connected graph $G$ can be obtained from any $\{1\}$- inverse of $G$ $(\cite[cite]{[\@@bibref{}{BG}{}{}]})$. We use $M^{(1)}$ to denote any $\{{1}\}$-inverse of a matrix $M$, and let $(M)_{uv}$ denote the $(u,v)$-entry of $M$.

Lemma 2.1 $(\cite[cite]{[\@@bibref{}{BSZHW}{}{}]})$ Let $G$ be a connected graph. Then

[TABLE]

Let $1_{n}$ denote the column vector of dimension $n$ with all the entries equal one. We will often use $1$ to denote all-ones column vector if the dimension can be read from the context.

Lemma 2.2 $(\cite[cite]{[\@@bibref{}{BYZhZH}{}{}]})$  For any graph, we have $L^{\#}_{G}1=0.$

Lemma 2.3 $(\cite[cite]{[\@@bibref{}{FZZh}{}{}]})$  Let

[TABLE]

be a nonsingular matrix. If $A$ and $D$ are nonsingular, then

[TABLE]

where $S=D-CA^{-1}B.$

Lemma 2.4 $(\cite[cite]{[\@@bibref{}{LZhB}{}{}]})$  Let $L$ be the Laplacian matrix a graph of order $n$. For any $a>0$, we have

[TABLE]

Lemma 2.5 $(\cite[cite]{[\@@bibref{}{SWZhB}{}{}]})$  Let $G$ be a connected graph on $n$ vertices. Then

[TABLE]

Lemma 2.6 $(\cite[cite]{[\@@bibref{}{QL}{}{}]})$  Let

[TABLE]

be the Laplacian matrix of a connected graph. If $D$ is nonsingular, then

[TABLE]

is a symmetric $\{1\}$-inverse of $L$, where $H=A-BD^{-1}B^{T}$.

3 The resistance distance and Kirchhoff index of $G[F;H_{v},l]$

Let $F$ be a connected graph with vertex set $\{u_{1},u_{2},...,u_{n}\}$. Let $H_{v}$ be a connected graph on $m+1$ vertices with a specified vertex $v$ and $V(H_{v})=\{v_{1},v_{2},...,v_{m},v\}$. Let $G=G[F;H_{v},l]$. Note that $G$ has $n(m+1)$ vertices. Let $deg(v)=l$, $1\leq l\leq m$. With loss of generality assume that $N(v)=\{v_{1},v_{2},...,v_{l}\}$. Let $H_{1}$ be the subgraph of $H_{v}$ induced by the vertices in $\{v_{1},v_{2},...,v_{l}\}$ and $H_{2}$ be the subgraph of $H_{v}$ induced by the vertices $\{v_{l+1},v_{l+2},...,v_{m}\}$. Suppose that $H_{v}=H_{1}\vee(H_{2}+\{v\})$. In this section, we focus on determing the resistance distance and Kirchhoff index of $G[F;H_{v},l]$ in terms of the resistance distance and Kirchhoff index of $F,H_{1}$ and $H_{2}$.

Theorem 3.1   Let $G[F;H_{v},l]$ be the graph as described above. Suppose that $H_{v}=H_{1}\vee(H_{2}+\{v\})$. Let the Laplacian spectrum of $H_{1}$ and $H_{2}$ be $\sigma(H_{1})=(0=\mu_{1},\mu_{2},\ldots\mu_{l})$ and $\sigma(H_{2})=(0=\nu_{1},\nu_{2},\ldots\nu_{m-l})$. Then $G[F;H_{v},l]$ have the resistance distance and Kirchhoff index as follows:

(i) For any $i,j\in V(F)$, we have

[TABLE]

(ii) For any $i\in V(F)$, $j\in V(H_{1})$, we have

[TABLE]

(iii) For any $i\in V(F)$, $j\in V(H_{2})$, we have

[TABLE]

(iv)For any $i\in V(H_{1})$, $j\in V(H_{2})$, we have

$r_{ij}(G[F;H_{v},l])$

[TABLE]

(v)For any $i\in V(H_{2})$, $j\in V(H_{1})$, we have

[TABLE]

(iv) $Kf(G[F;H_{v},l])$

[TABLE]

Proof  Let $v^{i}_{j}$ denote the $j$th vertex of $H_{v}$ in the $i$th copy of $H_{v}$ in $G$, for $i=1,2,\ldots,n$; $j=1,2,\ldots,m$, and let $V_{j}(H_{v})=\{v^{1}_{j},v^{2}_{j},\ldots,v^{n}_{j}\}$. Then $V(F)\cup(\cup_{j=1}^{m}V_{j}(H_{v}))$ is a partition of $V(G)$. Using this partition, the Laplacian matrix of $G=G[F;H_{v},l]$ can be expressed as

[TABLE]

We begin with the computation of $\{1\}$-inverse of $G=G[F;H_{v},l]$.

Let $A=L(F)+lI_{n}$, B=\left(\begin{array}[]{cccccccccccccccc}-1_{l}^{T}\otimes I_{n}&0\\ \end{array}\right), B^{T}=\left(\begin{array}[]{cccccccccccccccc}-1_{l}\otimes I_{n}&\\ 0&\\ \end{array}\right), and

[TABLE]

First we computer the $D^{-1}$. By Lemma 2.3, we have

[TABLE]

so $(A_{1}-B_{1}D_{1}^{-1}C_{1})^{-1}=[(L(H_{1})+(m-l+1)I_{l}-\frac{m-l}{l}J_{l\times l}]^{-1}\otimes I_{n}$.

By Lemma 2.3, we have

[TABLE]

By Lemma 2.3, we have

[TABLE]

Similarly, $-S^{-1}C_{1}A_{1}^{-1}=(-A_{1}^{-1}B_{1}S^{-1})^{T}=\frac{1}{l}J_{(m-l)\times l}\otimes I_{n}$. So

$D^{-1}$

[TABLE]

Now we compute the $\{1\}$-inverse of $G[F;H_{v},l]$. By Lemma 2.5, we have

[TABLE]

so $H^{\#}=L^{\#}(F)$.

According to Lemma 2.6, we calculate $-H^{\#}BD^{-1}$ and $-D^{-1}B^{T}H^{\#}$.

$-H^{\#}BD^{-1}$

[TABLE]

and

[TABLE]

We are ready to compute the $D^{-1}B^{T}H^{\#}BD^{-1}$.

[TABLE]

Let $P=[(L(H_{1})+(m-l+1)I_{l}-\frac{m-l}{l}J_{l\times l}]\otimes I_{n}$, $Q=(L(H_{2})+lI_{m-l}-\frac{l}{m-l+1}J_{(m-l)\times(m-l)})^{-1}\otimes I_{n}$, $M=\frac{1}{l}J_{l\times(m-l)}\otimes I_{n}+(1_{l}\otimes I_{n})L^{\#}(F)(1^{T}_{m-l}\otimes I_{n})$, then based on Lemma 2.6, the following matrix

$N=$

[TABLE]

is a symmetric $\{1\}$-inverse of $G[F;H_{v},l]$.

For any $i,j\in V(F)$, by Lemma 2.1 and the Equation $(3.1)$, we have

[TABLE]

as stated in (i).

For any $i\in V(F)$, $j\in V(H_{1})$, by Lemma 2.1 and the Equation $(3.1)$, we have

[TABLE]

as stated in (ii).

For any $i\in V(F)$, $j\in V(H_{2})$, by Lemma 2.1 and the Equation $(3.1)$, we have

[TABLE]

as stated in (iii).

For any $i\in V(H_{1})$, $j\in V(H_{2})$, by Lemma 2.1 and the Equation $(3.1)$, we have

[TABLE]

as stated in (iv).

For any $i\in V(H_{2})$, $j\in V(H_{1})$, by Lemma 2.1 and the Equation $(3.1)$, we have

[TABLE]

as stated in (v).

Now we compute the Kirchhoff index of $G[F;H_{v},l]$. By Lemma 2.5, we have

$Kf(G[F;H_{v},l])$

[TABLE]

Note that the eigenvalues of $(L(H_{2})+lI_{m-l})$ are $0+l,\nu_{2}(H_{2})+l,...,\nu_{m-l}(H_{2})+l$ and the eigenvalues of $J_{(m-l)\times(m-l)}$ is $(m-l),0^{(m-l-1)}$. Then

[TABLE]

[TABLE]

Similarly, $tr\left((L(H_{1})+(m-l+1)I_{l}-\frac{m-l}{l}J_{l\times l})^{-1}\otimes I_{n})\right)=n\sum_{i=2}^{l}\frac{1}{\mu_{i}(H_{1})+(m-l+1)}+n.$

It is easily obtained

[TABLE]

[TABLE]

[TABLE]

Let $P=\left(L(H_{1})+(m-l+1)I_{l}-\frac{m-l}{l}J_{l\times l}\right)\otimes I_{n}$, then

[TABLE]

[TABLE]

Let $Q=\left(L(H_{2})+lI_{m-l}-\frac{l}{m-l+1}J_{(m-l)\times(m-l)}\right)$, then

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Similarly, $1^{T}((1_{l}\otimes I_{n})L^{\#}(F)(1^{T}_{m-l}\otimes I_{n})1=0,1^{T}((1_{m-l}\otimes I_{n})L^{\#}(F)(1^{T}_{l}\otimes I_{n})1=0$ and $1^{T}((1_{m-l}\otimes I_{n})L^{\#}(F)(1^{T}_{l}\otimes I_{n})1=0$

Plugging $(3.2)$,$(3.3)$,$(3.4)$, $(3.5)$ and (3.6) and the above equation into $Kf(G[F;H_{v},l])$, we obtain the required result in (iv).

4 Resistance distance and Kirchhoff index of $G[F,u_{1},u_{2},\ldots,u_{k};H_{v},l]$

In this section, we consider the case when $F=F_{1}\vee F_{2}$, where $F_{1}$ the subgraph of $F$ induced by the vertices $u_{1},u_{2},\ldots,u_{k}$ and $F_{2}$ is the subgraph of $F$ induced by the vertices $u_{k+1},u_{k+2},\ldots,u_{n}$. In this case, we give the explicit formulate of resistance distance and Kirchhoff index of $G=G[F,u_{1},u_{2},\ldots,u_{k};H_{v},l]$.

Theorem 4.1  Let $G=G[F,u_{1},u_{2},\ldots,u_{k};H_{v},l]$ be the graph as described above. Let $\sigma(F_{1})=(0=\alpha_{1},\alpha_{2},\ldots\alpha_{k})$, $\sigma(F_{2})=(0=\beta_{1},\beta_{2},\ldots\beta_{n-k})$ and $\sigma(H_{1})=(0=\mu_{1},\mu_{2},\ldots\mu_{l})$, and $\sigma(H_{2})=(0=\nu_{1},\nu_{2},\ldots\nu_{m-l})$. Then $G$ have the resistance distance and Kirchhoff index as follows:

(i)For any $i,j\in V(F_{1})$, we have

[TABLE]

(ii)For any $i,j\in V(F_{2})$, we have

[TABLE]

(iii)For any $i,j\in V(H_{1})$, we have

[TABLE]

(iv)For any $i,j\in V(H_{2})$, we have

[TABLE]

(v)For any $i\in V(F)$,$j\in V(H_{1})$, we have

[TABLE]

(vi)For any $i\in V(F)$, $j\in V(H_{2})$, we have

[TABLE]

(vii)For any $i\in V(H_{1})$, $j\in V(H_{2})$, we have

[TABLE]

(viii)For any $i\in V(H_{2})$, $j\in V(H_{1})$, we have

[TABLE]

(ix)

[TABLE]

Proof  Let $v^{i}_{j}$ denote the $j$th vertex of $H_{v}$ in the $i$th copy of $H_{v}$ in $G$, for $i=1,2,\ldots,k$; $j=1,2,\ldots,m$, and let $V_{j}(H_{v})=\{v^{1}_{j},v^{2}_{j},\ldots,v^{k}_{j}\}$. Then $V(F_{1})\cup V(F_{2})\cup(\cup_{j=1}^{m}V_{j}(H_{v}))$ is a partition of the vertex set of $G=G[F,u_{1},u_{2},\ldots,u_{k};H_{v},l]$. Using this partition, the Laplacian matrix of $G$ can be expressed as

[TABLE]

where $L_{1}=L(F_{1})+(n-k+l)I_{k}$, $L_{2}=L(F_{2})+kI_{n-k}$, $L_{3}=(L(H_{1})+(m-l+1)I_{l})\otimes I_{k}$, and $L_{4}=(L(H_{2})+lI_{m-l})\otimes I_{k}.$

Let $A=L_{1}$, B=\left(\begin{array}[]{ccccccc}-J_{k\times(n-k)}&-1^{T}_{l}\otimes I_{k}&0\\ \end{array}\right), B^{T}=\left(\begin{array}[]{ccccccc}-J_{(n-k)\times k}&\\ -1_{l}\otimes I_{k}&\\ 0&\\ \end{array}\right), and

[TABLE]

First, we compute D_{1}^{-1}=\left(\begin{array}[]{cccccccccccccccc}L_{3}&-J_{l\times(m-l)}\otimes I_{k}\\ -J_{(m-l)\times l}\otimes I_{k}&L_{4}\\ \end{array}\right)^{-1}. By Lemma 2.3, we have

[TABLE]

so $(A_{1}-B_{1}D_{1}^{-1}C_{1})^{-1}=[(L(H_{1})+(m-l+1)I_{l}-\frac{m-l}{l}J_{l\times l}]^{-1}\otimes I_{k}$.

By Lemma 2.3, we have

[TABLE]

By Lemma 2.3, we have

[TABLE]

Similarly, $-S^{-1}C_{1}A_{1}^{-1}=(-A_{1}^{-1}B_{1}S^{-1})^{T}=\frac{1}{l}J_{(m-l)\times l}\otimes I_{k}$. So

$D_{1}^{-1}=$

[TABLE]

Now we compute the $\{1\}$-inverse of $G[F,u_{1},u_{2},\ldots,u_{k};H_{v},l]$. Let $P=[(L(H_{1})+(m-l+1)I_{l}-\frac{m-l}{l}J_{l\times l}]\otimes I_{k}$, $Q=[(L(H_{2})+lI_{m-l}-\frac{l}{m-l+1}J_{(m-l)\times(m-l)})]\otimes I_{k}$. By Lemma 2.6, we have

[TABLE]

so $H^{\#}=(L(F_{1})+(n-k)I_{k}-\frac{n-k}{k}J_{k\times k})^{\#}$. By Lemmma 2.4, $H^{\#}=(L(F_{1})+(n-k)I_{k})^{-1}-\frac{1}{k(n-k)}J_{k\times k}$.

According to Lemma 2.6, we calculate $-H^{\#}BD^{-1}$ and $-D^{-1}B^{T}H^{\#}$.

$-H^{\#}BD^{-1}$

[TABLE]

and

[TABLE]

We are ready to compute the $D^{-1}B^{T}H^{\#}BD^{-1}$.

$D^{-1}B^{T}H^{\#}BD^{-1}$

[TABLE]

Let $M=1^{T}_{m-l}\otimes I_{k}$, $N=1^{T}_{l}\otimes I_{k}$. Based on Lemma 2.6, the following matrix

[TABLE]

is a symmetric $\{1\}$-inverse of $G=G[F,u_{1},u_{2},\ldots,u_{k};H_{v},l]$, $P=[(L(H_{1})+(m-l+1)I_{l}-\frac{m-l}{l}J_{l\times l}]\otimes I_{k}$, $Q=[(L(H_{2})+lI_{m-l}-\frac{l}{m-l+1}J_{(m-l)\times(m-l)})]\otimes I_{k}$.

For any $i,j\in V(F_{1})$, by Lemma 2.1 and the Equation $(4.7)$, we have

[TABLE]

as stated in (i).

For any $i,j\in V(F_{2})$, by Lemma 2.1 and the Equation $(4.7)$, we have

[TABLE]

as stated in (ii).

For any $i,j\in V(H_{1})$, by Lemma 2.1 and the Equation $(4.7)$, we have

[TABLE]

as stated in (iii).

For any $i,j\in V(H_{2})$, by Lemma 2.1 and the Equation $(4.7)$, we have

[TABLE]

as stated in (iv).

For any $i\in V(F)$,$j\in V(H_{1})$ by Lemma 2.1 and the Equation $(4.7)$, we have

[TABLE]

as stated in (v).

For any $i\in V(F)$, $j\in V(H_{2})$, by Lemma 2.1 and the Equation $(4.7)$, we have

[TABLE]

as stated in (vi).

For any $i\in V(H_{1})$, $j\in V(H_{2})$, by Lemma 2.1 and the Equation $(4.7)$, we have

[TABLE]

as stated in (vii).

For any $i\in V(H_{2})$, $j\in V(H_{1})$, by Lemma 2.1 and the Equation $(4.7)$, we have

[TABLE]

as stated in (viii).

Now we compute the Kirchhoff index of $G[F,u_{1},u_{2},\ldots,u_{k};H_{v},l]$.

$Kf(G[F,u_{1},u_{2},\ldots,u_{k};H_{v},l])$

[TABLE]

Note that the eigenvalues of $(L(F_{1})+(n-k)I_{k})$ are $\alpha_{1}+(n-k),\alpha_{2}+(n-k),...,\alpha_{k}+(n-k)$. Then

[TABLE]

Similarly, $tr\left((L(F_{2})+kI_{n-k})^{-1}\right)=\sum_{i=1}^{n-k}\frac{1}{\beta_{i}+k}.$

Note that the eigenvalues of $(L(H_{1})+(m-l+1)I_{l}$ are $0+(m-l+1),\mu_{2}(H_{1})+(m-l+1),...,\mu_{l}(H_{1})+(m-l+1)$ and the eigenvalues of $J_{(m-l)\times(m-l)}$ is $(m-l),0^{(m-l-1)}$. Then

[TABLE]

[TABLE]

Similarly,

[TABLE]

It is easily obtained that $tr(J_{l\times(m-l)}\otimes I_{k})=lk,tr(J_{(m-l)\times l}\otimes I_{k})=(m-l)k,$ and $tr(N^{T}H^{\#}N)+tr(M^{T}H^{\#}M)=tr(J_{l\times l}\otimes H^{\#})+tr(J_{(m-l)\times(m-l)}\otimes H^{\#})=ltr(H^{\#})+(m-l)tr(H^{\#})=mtr(H^{\#})$.

Since $1^{T}_{k}H^{\#}=1^{T}_{k}[(L(F_{1})+(n-k)I_{k})^{-1}-\frac{1}{k(n-k)}J_{k\times k}]=\frac{1}{n-k}1^{T}_{k}-\frac{1}{k(n-k)}1^{T}_{k}=0$, then

[TABLE]

By the process of Theorem 3.1, we have

[TABLE]

[TABLE]

Similarly, $1^{T}(M^{T}H^{\#}M)1=0$, $1^{T}N^{T}H^{\#}M1=0$ and $1^{T}M^{T}H^{\#}N1=0$.

[TABLE]

Similarly, $1^{T}(J_{(m-l)\times l}\otimes I_{k})=lk(m-l)$.

Plugging the above equation into $Kf(G[F,u_{1},u_{2},\ldots,u_{k};H_{v},l])$, we obtain the required result in (iv).

5 Conclusion

In this paper, using the Laplacian generalized inverse approach, we obtained the resistance distance and Kirchhoff indices of $G=G[F,u_{1},...,u_{k},H_{v}]$ in terms of the resistance distance and Kirchhoff index $F$ and $H_{uv}$, respectively.

This article has been reviewed in Applied Mathematics-A Journal of Chinese Universities on September 26, 2018.

Acknowledgment: This work was supported by the National Natural Science Foundation of China (Nos. 11461020), the Research Foundation of the Higher Education Institutions of Gansu Province, China (2018A-093), the Science and Technology Plan of Gansu Province (18JR3RG206) and Research and Innovation Fund Project of President of Hexi University (XZZD2018003).