On distance matrices of graphs

TL;DR

This paper extends the understanding of distance matrices in graphs by deriving formulas for their inverses, generalizing known results for trees and cactoid digraphs through the theory of Laplacian expressible matrices.

Contribution

It introduces a novel approach to compute the inverse of distance matrices in complex graphs using Laplacian expressible matrices, generalizing previous results.

Findings

Explicit formulas for the inverse of the distance matrix of weighted cactoid digraphs.

Extension of Graham, Hoffman, and Hosoya's determinant formula to inverse matrices.

Connection of the results to known formulas for trees and specific graph classes.

Abstract

Distance well-defined graphs consist of connected undirected graphs, strongly connected directed graphs and strongly connected mixed graphs. Let $G$ be a distance well-defined graph, and let ${\sf D}(G)$ be the distance matrix of $G$. Graham, Hoffman and Hosoya [3] showed a very attractive theorem, expressing the determinant of ${\sf D}(G)$ explicitly as a function of blocks of $G$. In this paper, we study the inverse of ${\sf D}(G)$ and get an analogous theory, expressing the inverse of ${\sf D}(G)$ through the inverses of distance matrices of blocks of $G$ (see Theorem 3.3) by the theory of Laplacian expressible matrices which was first defined by the first author [9]. A weighted cactoid digraph is a strongly connected directed graph whose blocks are weighted directed cycles. As an application of above theory, we give the determinant and the inverse of the distance matrix of a weighted cactoid digraph, which imply Graham and Pollak's formula and the inverse of the distance matrix of a tree.

Equations

$$n-1=\sum _{i=1}^r(n_i-1).$$

$$\mathsf{bi}(G)=r-1.$$

$$n+\mathsf{bi}(G)=\sum _{i=1}^r{n}_i.$$

$$\lambda$$

$${\alpha }_v$$

$${\beta }_v$$

$$L$$

$${\alpha }_v-1$$

$${\beta }_v-1$$

$${\alpha }^T\mathbf{j}=\sum _{v\in V(G)}{\alpha }_v$$

$$D_F$$

$$D_H$$

$$D=\left(\begin{array}[]{ccc}0&a^{T}&b^{T}\\ c&D_{F,x_{0}}&M\\ d&N&D_{H,x_{0}}\end{array}\right),$$

$${\mathsf{BI}}_G(x_0)$$

$${\mathsf{BI}}_G(x_i)$$

$${\mathsf{BI}}_G(x_{p+j})$$

$${\alpha }_{x_0}=({\alpha }_F{)}_{x_0}+({\alpha }_H{)}_{x_0}-1;$$

$${\alpha }_{x_i}=({\alpha }_F{)}_{x_i},$$

$${\alpha }_{x_{p+j}}=({\alpha }_H{)}_{x_{p+j}},$$

$${\hat{\alpha }}_F^T=[{\alpha }_{x_1},{\alpha }_{x_2},\cdots ,{\alpha }_{x_p}],$$

$${\hat{\alpha }}_H^T=[{\alpha }_{x_{p+1}},{\alpha }_{x_{p+2}},\cdots ,{\alpha }_{x_{p+q}}],$$

$${\alpha }^T=[{\alpha }_{x_0},{\hat{\alpha }}_F^T,{\hat{\alpha }}_H^T],$$

$${\alpha }_F^T=[({\alpha }_F{)}_{x_0},{\hat{\alpha }}_F^T],$$

$${\alpha }_H^T=[({\alpha }_H{)}_{x_0},{\hat{\alpha }}_H^T],$$

$${\gamma }_{x_0}={\alpha }^{T}D[\cdot |x_0]$$

$${\gamma }_v$$

$${\gamma }_w$$

$$L_F$$

$$L_H$$

$$L=\left(\begin{array}[]{ccc}\pi&f^{T}&g^{T}\\ h&L_{F,x_{0}}&Q\\ k&Q^{T}&L_{H,x_{0}}\end{array}\right),$$

$$L_F{D}_F+I$$

$$L_H{D}_H+I$$

$$LD+I$$

$$A_{11}$$

$$A_{12}$$

$$A_{13}$$

$$A_{21}$$

$$A_{22}$$

Full text

Keywords: Distance matrix; Determinant; Inverse matrix; Weighted cactoid digraph.

MSC: 15A15, 05C05.

1 Introduction

Let $G$ be a distance well-defined graph. We use $V(G)$ and $E(G)$ to denote the vertex set and edge (arc) set of $G$ respectively. The distance $\partial_{G}(u,v)$ from vertex $u$ to vertex $v$ in $G$ is the length (number of (directed) edges) of the shortest (directed) path from $u$ to $v$ in $G$. The distance matrix ${\sf D}(G)$ of $G$ is a $|V(G)|\times|V(G)|$ square matrix whose $(u,v)$-entry is the distance $\partial_{G}(u,v)$, that is ${\sf D}(G)=(\partial_{G}(u,v))_{u,v\in V(G)}$. A cut vertex of $G$ is a vertex whose deletion results in a disconnected graph. A block of $G$ is a distance well-defined subgraph on at least two vertices such that it has no cut vertices and is maximal with respect to this property.

Let ${\mathbf{j}}$ be an appropriate size column vector whose entries are ones, let $\mathbf{0}$ be an appropriate size matrix whose entries are zeroes, and let $I$ be the identity matrix with an appropriate size. For any matrix $M$ and any column vector $\alpha$, we use $M^{T}$ and $\alpha^{T}$ to denote their transposes, respectively.

A square matrix $L$ is called a Laplacian-like matrix if $L{\mathbf{j}}=\mathbf{0}$ and ${\mathbf{j}}^{T}L=\mathbf{0}$. This definition was first defined by the first author [9] to generalize the Laplacian matrix of an undirected graph. To study the inverse of the distance matrix of a distance well-defined graph, the first author [9] gave the definition of Laplacian expressible matrices. A square matrix $D$ is left (or right) Laplacian expressible if there exist a number $\lambda$, a column vector $\beta$ with $\beta^{T}{\mathbf{j}}=1$, and a square matrix $L$ such that

$L{\mathbf{j}}=\mathbf{0}$, $\beta^{T}D=\lambda{\mathbf{j}}^{T}$ and $LD+I=\beta{\mathbf{j}}^{T}$

(or ${\mathbf{j}}^{T}L=\mathbf{0}$, $D\beta=\lambda{\mathbf{j}}$ and $DL+I={\mathbf{j}}\beta^{T}$).

If the number $\lambda\neq 0$, then the Laplacian expressible matrix $D$ is invertible with $D^{-1}=-L+\frac{1}{\lambda}\beta\beta^{T}$ (see Lemma 2.1) and the matrix $L$ is Laplacian-like (see Lemma 2.2), which means the inverse $D^{-1}$ is expressed as the sum of a Laplacian-like matrix and a rank one matrix. This is why “Laplacian expressible” comes. The main theory of [9] shows that if the distance matrix of each block of a graph is left (or right) Laplacian expressible, then the distance matrix of the graph is also left (or right) Laplacian expressible. This theory is very helpful for us to calculate the inverse of the distance matrix of a graph whose blocks correspond to Laplacian expressible matrices.

In this paper, we investigate the inverse of the distance matrix of a graph further and give a generalization of the Laplacian expressible theory in [9]. We give the following crucial definition (see also Definition 2.3). A square matrix $D$ is modified left (or right) Laplacian expressible if there exist a number $\lambda$, column vectors $\alpha$ and $\beta$, and a square matrix $L$ such that

$\alpha^{T}{\mathbf{j}}=1$, $L{\mathbf{j}}=\mathbf{0}$, $\alpha^{T}D=\lambda{\mathbf{j}}^{T}$ and $LD+I=\beta{\mathbf{j}}^{T}$

(or ${\mathbf{j}}^{T}\beta=1$, ${\mathbf{j}}^{T}L=\mathbf{0}$, $D\beta=\lambda{\mathbf{j}}$ and $DL+I={\mathbf{j}}\alpha^{T}$).

By definition, we know the Laplacian expressible matrix is a special case of the modified Laplacian expressible matrix (when $\alpha=\beta$). Let $D$ be a modified left (or right) Laplacian expressible matrix as above. Suppose $\lambda\neq 0$ and {\mathbf{j}}^{T}\beta=1\text{ (or \alpha^{T}{\mathbf{j}}=1)}, then by Lemmas 2.1 and 2.2, the matrix $D$ is invertible, $L$ is Laplacian-like, and the inverse $D^{-1}=-L+\frac{1}{\lambda}\beta\alpha^{T}$ is expressed as the sum of a Laplacian-like matrix and a rank one matrix.

If the distance matrices of blocks of a graph $G$ are modified left (or right) Laplacian expressible, then we get the distance matrix of $G$ is also modified left (or right) Laplacian expressible (see Theorem 3.3). So this modified Laplacian expressible property of a graph can retain from its blocks. This is helpful for us to calculate the inverse of the distance matrix of a graph whose blocks have modified Laplacian expressible distance matrices. This theory generalizes results in [9], which first used Laplacian expressible matrices and got the inverses of distance matrices of the following graphs: trees, weighted trees, block graphs, odd-cycle-clique graphs, bi-block graphs, cactoid digraphs and complete multipartite graphs under some condition, etc (references are therein). But our theory can do more, such as arc weighted trees [2, 8], mixed block graphs [7], weighted cactoid digraphs (see Theorem 4.3), etc. Theorem 4.3 implies the corresponding results of weighted trees [2, 8] and cactoid digraphs [6]. For each of the above graphs, the inverse of its distance matrix can be expressed as the sum of a Laplacian-like matrix and a rank one matrix.

2 The modified Laplacian expressible matrix

First, we give a formula on the inverse of a square matrix.

Lemma 2.1

Let $D$ be an $n\times n$ matrix. Let $\lambda$ be an nonzero number, let $\alpha$ and $\beta$ be $n\times 1$ column vectors, and let $L$ be an $n\times n$ matrix. If either

$\alpha^{T}D=\lambda{\mathbf{j}}^{T}$* and $LD+I=\beta{\mathbf{j}}^{T}$, or* 2. 2.

$D\beta=\lambda{\mathbf{j}}$* and $DL+I={\mathbf{j}}\alpha^{T}$,*

then $D$ is invertible and $D^{-1}=-L+\frac{1}{\lambda}\beta\alpha^{T}$.

• Proof.  

The proofs of the two cases are similar, so we only give the proof of the first case. Suppose $\alpha^{T}D=\lambda{\mathbf{j}}^{T}$ and $LD+I=\beta{\mathbf{j}}^{T}$. Then $\beta\alpha^{T}D=\lambda\beta{\mathbf{j}}^{T}$ and $\beta{\mathbf{j}}^{T}=\frac{1}{\lambda}\beta\alpha^{T}D$. So $LD+I=\frac{1}{\lambda}\beta\alpha^{T}D$ and $(-L+\frac{1}{\lambda}\beta\alpha^{T})D=I$. Thus $D^{-1}=-L+\frac{1}{\lambda}\beta\alpha^{T}$. $\blacksquare$

Now we give some sufficient conditions for a square matrix $L$ to be a Laplacian-like matrix.

Lemma 2.2

Let $D$ be an $n\times n$ invertible matrix. Let $\lambda$ be an nonzero number, let $\alpha$ and $\beta$ be $n\times 1$ column vectors, and let $L=-D^{-1}+\frac{1}{\lambda}\beta\alpha^{T}$. Then $L$ is a Laplacian-like matrix if one of the following conditions holds:

$\alpha^{T}D=\lambda{\mathbf{j}}^{T}$, ${\mathbf{j}}^{T}\beta=1$ and $L{\mathbf{j}}=\mathbf{0}$, 2. 2.

$D\beta=\lambda{\mathbf{j}}$, $\alpha^{T}{\mathbf{j}}=1$ and ${\mathbf{j}}^{T}L=\mathbf{0}$, 3. 3.

$\alpha^{T}D=\lambda{\mathbf{j}}^{T}$, $D\beta=\lambda{\mathbf{j}}$ and ${\mathbf{j}}^{T}\beta=\alpha^{T}{\mathbf{j}}=1$.

• Proof.  

The proofs are similar, so we only give the proof of the second case. Since $D\beta=\lambda{\mathbf{j}}$, we have $D^{-1}{\mathbf{j}}=\frac{1}{\lambda}\beta$. Thus $L{\mathbf{j}}=-D^{-1}{\mathbf{j}}+\frac{1}{\lambda}\beta\alpha^{T}{\mathbf{j}}=-\frac{1}{\lambda}\beta+\frac{1}{\lambda}\beta=\mathbf{0}$. Hence $L$ is a Laplacian-like matrix. $\blacksquare$

For ease of reference, we repeat the following definition.

Definition 2.3** (Modified Laplacian expressible matrix)**

Let $D$ be an $n\times n$ matrix. Let $\lambda$ be a number, let $\alpha$ and $\beta$ be $n\times 1$ column vectors, and let $L$ be an $n\times n$ matrix. If

$\alpha^{T}{\mathbf{j}}=1$, $L{\mathbf{j}}=\mathbf{0}$, $\alpha^{T}D=\lambda{\mathbf{j}}^{T}$ and $LD+I=\beta{\mathbf{j}}^{T}$,

then we call $D$ a modified left Laplacian expressible matrix, or

a left ${\sf LapExp}^{*}(\lambda,\alpha,\beta,L)$ matrix

to specify the corresponding parameters $\lambda,\alpha,\beta,L$. If

${\mathbf{j}}^{T}\beta=1$, ${\mathbf{j}}^{T}L=\mathbf{0}$, $D\beta=\lambda{\mathbf{j}}$ and $DL+I={\mathbf{j}}\alpha^{T}$,

then we call $D$ a modified right Laplacian expressible matrix, or

a right ${\sf LapExp}^{*}(\lambda,\alpha,\beta,L)$ matrix

to specify the corresponding parameters $\lambda,\alpha,\beta,L$. The matrix $D$ is called a modified Laplacian expressible matrix, if either $D$ is a modified left Laplacian expressible matrix, or $D$ is a modified right Laplacian expressible matrix.

Remark 2.4

The above Definition 2.3 is also suitable and the above lemmas in this section are also true if $\lambda$ and the entries of vectors and matrices are taken from a commutative ring with identity. In this case, the condition “$\lambda\neq 0$” should be changed to “$\lambda$ is invertible”.

Let $D$ be a left (or right) ${\sf LapExp}^{*}(\lambda,\alpha,\beta,L)$ matrix, and suppose $\lambda\neq 0$. Then by Lemma 2.1, the matrix $D$ is invertible and $D^{-1}=-L+\frac{1}{\lambda}\beta\alpha^{T}$. If we assume ${\mathbf{j}}^{T}\beta=1$ (or $\alpha^{T}{\mathbf{j}}=1$) additionally, then by Lemma 2.2, the matrix $L$ is a Laplacian-like matrix, and so the inverse matrix $D^{-1}$ can be expressed as the sum of a Laplacian-like matrix and a rank one matrix. This is why we call the matrix $D$ a modified Laplacian expressible matrix.

3 Inverse of the generalized distance matrix

Definition 3.1** (Generalized distance matrix)**

Let $G$ be a distance well-defined graph. A generalized distance matrix $D$ of $G$ is a $|V(G)|\times|V(G)|$ matrix $(D_{uv})_{u,v\in V(G)}$ whose entries are taken from a commutative ring with identity and satisfy the following conditions:

$D_{uu}=0$* for all $u\in V(G)$, and* 2. 2.

if $u$ and $v$ are two vertices of $G$ such that every shortest (directed) path from $u$ to $v$ passes through the cut-vertex $x$, then $D_{uv}=D_{ux}+D_{xv}$.

Let $G$ be a distance well-defined graph with $n$ vertices. The distance matrix of $G$ is actually a generalized distance matrix. Let $G_{1},G_{2},\ldots,G_{r}$ be all the blocks of $G$ where $r\geqslant 1$. For each $1\leqslant i\leqslant r$, let $n_{i}\geqslant 2$ be the number of vertices of $G_{i}$. We call $(G^{n};G_{1}^{n_{1}},G_{2}^{n_{2}},\ldots,G_{r}^{n_{r}})$ the structure parameters of $G$. Note that

[TABLE]

An $n$-bag is a tuple $(A,\lambda,\alpha,\beta,L)$ consisting of $n\times n$ matrices $A$ and $L$, a number $\lambda$ and $n\times 1$ column vectors $\alpha$ and $\beta$. An $n$-bag $(A,\lambda,\alpha,\beta,L)$ is called a left (or right) ${\sf LapExp}^{*}$ $n$-bag if $A$ is a left (or right) ${\sf LapExp}^{*}(\lambda,\alpha,\beta,L)$ matrix. Let $M$ be an $n\times n$ matrix whose rows and columns are indexed by vertices of $G$, and let $H$ be a subgraph of $G$. We use ${\sf sub}(M;G,H)$ to denote the submatrix of $M$ whose rows and columns are corresponding to the vertices of $H$. Let $v$ be a vertex of $G$. The block index set ${\sf BI}_{G}(v)$ of $v$ is the set of all indices $k$, $1\leqslant k\leqslant r$, satisfying that $v$ is a vertex of the block $G_{k}$. The block index of $v$ is the cardinality ${\sf bi}_{G}(v)=\#{\sf BI}_{G}(v)$. The block index of $G$ is ${\sf bi}(G)=\sum\limits_{v\in V(G)}({\sf bi}_{G}(v)-1)$. By Lemma 4.2 in [9], we have

[TABLE]

So

[TABLE]

Definition 3.2** (Composition bag)**

Let $G$ be a distance well-defined graph with structure parameters $(G^{n};G_{1}^{n_{1}},G_{2}^{n_{2}},\ldots,G_{r}^{n_{r}})$. Let $D$ be a generalized distance matrix of $G$. For each $1\leqslant i\leqslant r$, let $D_{i}={\sf sub}(D;G,G_{i})$ and let $B_{i}=(D_{i},\lambda_{i},\alpha_{i},\beta_{i},L_{i})$ be an $n_{i}$-bag. The composition bag of bags $B_{1},B_{2},\ldots,B_{r}$ is an $n$-bag $(D,\lambda,\alpha,\beta,L)$ whose parameters are defined as follows:

[TABLE]

where

for any vertex $v$ of $G$, the entry of $\alpha$ corresponding to $v$ is $\alpha_{v}$ and the entry of $\beta$ corresponding to $v$ is $\beta_{v}$, and 2. 2.

for each $1\leqslant i\leqslant r$, $\hat{L_{i}}$ is an $n\times n$ matrix such that the entry of $\hat{L_{i}}$ corresponding to vertices $u$ and $v$ of $G$ is

(\hat{L_{i}})_{uv}=\left\{\begin{array}[]{cl}(L_{i})_{uv},&\text{if }u,v\in V(G_{i}),\\ 0,&\text{otherwise}.\end{array}\right.

Note that in the above definition, for any vertex $v$ of $G$, we can write

[TABLE]

this means $\alpha_{v}$ (or $\beta_{v}$) depends only on entries $(\alpha_{i})_{v}$ \Bigl{(}or (\beta_{i})_{v}\Bigr{)} with $i\in{\sf BI}_{G}(v)$.

In general, we get the following theorem on the generalized distance matrix of a distance well-defined graph whose blocks are corresponding to (left or right) ${\sf LapExp}^{*}$ bags.

Theorem 3.3

Let $G$ be a distance well-defined graph with structure parameters $(G^{n};G_{1}^{n_{1}},G_{2}^{n_{2}},\ldots,G_{r}^{n_{r}})$. Let $D$ be a generalized distance matrix of $G$. For each $1\leqslant i\leqslant r$, let $D_{i}={\sf sub}(D;G,G_{i})$ and let $B_{i}=(D_{i},\lambda_{i},\alpha_{i},\beta_{i},L_{i})$ be a left (or right) ${\sf LapExp}^{*}$ $n_{i}$-bag. Let $B=(D,\lambda,\alpha,\beta,L)$ be the composition bag of bags $B_{1},B_{2},\ldots,B_{r}$. Then

$\alpha^{T}{\mathbf{j}}=1$*, $L{\mathbf{j}}=\mathbf{0}$, $\alpha^{T}D=\lambda{\mathbf{j}}^{T}$ and *$LD+I=\beta{\mathbf{j}}^{T}$

(or ${\mathbf{j}}^{T}\beta=1$, ${\mathbf{j}}^{T}L=\mathbf{0}$, $D\beta=\lambda{\mathbf{j}}$ and $DL+I={\mathbf{j}}\alpha^{T}$).

As a consequence, $B=(D,\lambda,\alpha,\beta,L)$ is a left (or right) ${\sf LapExp}^{*}$ $n$-bag. Furthermore, if $\lambda$ is invertible, then $D$ is invertible and $D^{-1}=-L+\frac{1}{\lambda}\beta\alpha^{T}$.

• Proof.  

The proofs of left and right cases are similar. Here we only give the proof of the left case. For each $1\leqslant i\leqslant r$, the bag $B_{i}=(D_{i},\lambda_{i},\alpha_{i},\beta_{i},L_{i})$ is a left ${\sf LapExp}^{*}$ $n_{i}$-bag, i.e.

$\alpha_{i}^{T}{\mathbf{j}}=1$, $L_{i}{\mathbf{j}}=\mathbf{0}$, $\alpha_{i}^{T}D_{i}=\lambda_{i}{\mathbf{j}}^{T}$ and $L_{i}D_{i}+I=\beta_{i}{\mathbf{j}}^{T}$.

By counting in two ways, we have

$r=\sum\limits_{i=1}^{r}\alpha_{i}^{T}{\mathbf{j}}=\sum\limits_{i=1}^{r}\sum\limits_{v\in V(G_{i})}(\alpha_{i})_{v}=\sum\limits_{v\in V(G)}\sum\limits_{i\in{\sf BI}_{G}(v)}(\alpha_{i})_{v}$.

Then by Equation (1), we get

[TABLE]

For each $1\leqslant i\leqslant r$, since $L_{i}{\mathbf{j}}=\mathbf{0}$, we get $\hat{L_{i}}{\mathbf{j}}=\mathbf{0}$. By definition, we have $L{\mathbf{j}}=\sum\limits_{i=1}^{r}\hat{L_{i}}{\mathbf{j}}=\mathbf{0}$.

We will prove $\alpha^{T}D=\lambda{\mathbf{j}}^{T}$ by induction on the block number $r$. The case $r=1$ is clear. Now let $r\geqslant 2$. Without loss of generality, we may assume $H=G_{r}$ is a leaf block of $G$ with separating vertex $x_{0}$. Let $B_{H}=(D_{H},\lambda_{H},\alpha_{H},\beta_{H},L_{H})=B_{r}$. Then $\alpha_{H}^{T}{\mathbf{j}}=1$ and $\alpha_{H}^{T}D_{H}=\lambda_{H}{\mathbf{j}}^{T}$. Let $F=G-(H-x_{0})$. Then the blocks of $F$ are $G_{1},G_{2},\ldots,G_{r-1}$. Let $D_{F}={\sf sub}(D;G,F)$. Then $D_{F}$ is a generalized distance matrix of $F$, and $D_{i}={\sf sub}(D_{F};F,G_{i})$ for $1\leqslant i\leqslant r-1$. Let $B_{F}=(D_{F},\lambda_{F},\alpha_{F},\beta_{F},L_{F})$ be the composition bag of $B_{1},\ldots,B_{r-1}$. Then $\alpha_{F}^{T}{\mathbf{j}}=1$ by the first paragraph of this proof, and $\alpha_{F}^{T}D_{F}=\lambda_{F}{\mathbf{j}}^{T}$ by the induction hypothesis.

Assume the vertex set of $F$ and $H$ are $V(F)=\{x_{0},x_{1},x_{2},\ldots,x_{p}\}$ and $V(H)=\{x_{0},x_{p+1},x_{p+2},\ldots,x_{p+q}\}$, respectively. Then the vertex set of $G$ is $V(G)=\{x_{0},x_{1},x_{2},\ldots,x_{p},x_{p+1},\ldots,x_{p+q}\}$. The matrices $D_{F}=(d_{uv})_{u,v\in V(F)}$ and $D_{H}=(d_{uv})_{u,v\in V(H)}$ are generalized distance matrices of $F$ and $H$, respectively. We can write them in the following form

[TABLE]

where $a^{T}=[a_{1},\cdots,a_{p}]$, $b^{T}=[b_{1},\cdots,b_{q}]$, $c^{T}=[c_{1},\cdots,c_{p}]$, $d^{T}=[d_{1},\cdots,d_{q}]$, $D_{F,x_{0}}=(d_{x_{i}x_{j}})_{1\leqslant i,j\leqslant p}$ and $D_{H,x_{0}}=(d_{x_{p+i}x_{p+j}})_{1\leqslant i,j\leqslant q}$. Then by the definition of the generalized distance matrix, we have

[TABLE]

where $M=(m_{ij})_{p\times q}$ with $m_{ij}=c_{i}+b_{j}$ for $1\leqslant i\leqslant p$ and $1\leqslant j\leqslant q$, and $N=(n_{st})_{q\times p}$ with $n_{st}=d_{s}+a_{t}$ for $1\leqslant s\leqslant q$ and $1\leqslant t\leqslant p$. Then $M=c{\mathbf{j}}^{T}+{\mathbf{j}}b^{T}$ and $N=d{\mathbf{j}}^{T}+{\mathbf{j}}a^{T}$. Note that

[TABLE]

Then

[TABLE]

Let

[TABLE]

So we can write

[TABLE]

Let $\gamma^{T}=\alpha^{T}D$. For the entry corresponding to the vertex $x_{0}$,

[TABLE]

Let $v$ be a vertex of $F-x_{0}$. Then $N[\cdot|v]=d+a_{v}{\mathbf{j}}$ and

[TABLE]

Let $w$ be a vertex of $H-x_{0}$. A similar argument gives $\gamma_{w}=\lambda$. Now we have $M[\cdot|w]=c+b_{w}{\mathbf{j}}$. Then

[TABLE]

By the above calculation, we have $\gamma^{T}=\lambda{\mathbf{j}}^{T}$.

At the end of this proof, we will prove $LD+I=\beta{\mathbf{j}}^{T}$ by induction on the block number $r$. The case $r=1$ is clear. Now let $r\geqslant 2$, and we use notation as above. Then $L_{H}{\mathbf{j}}=\mathbf{0}$ and $L_{H}D_{H}+I=\beta_{H}{\mathbf{j}}^{T}$. For the subgraph $F$, we have $L_{F}{\mathbf{j}}=\mathbf{0}$ by the first paragraph of this proof, and $L_{F}D_{F}+I=\beta_{F}{\mathbf{j}}^{T}$ by the induction hypothesis. The matrices $L_{F}$ and $L_{H}$ can be written as

[TABLE]

where $f^{T}=[f_{1},\cdots,f_{p}]$, $g^{T}=[g_{1},\cdots,g_{q}]$, $h^{T}=[h_{1},\cdots,h_{p}]$ and $k^{T}=[k_{1},\cdots,k_{q}]$. Then by the definition of $L$, we have

[TABLE]

where $\pi=\varphi+\tau$, and $Q_{p\times q}=\mathbf{0}$. Assume

[TABLE]

Now we calculate the entries of $LD+I$. For the entries of the row corresponding to the vertex $x_{0}$, we have

[TABLE]

[TABLE]

[TABLE]

For the entries of the rows corresponding to the vertices of $F-x_{0}$, we have

[TABLE]

For the entries of the rows corresponding to the vertices of $H-x_{0}$, we have

[TABLE]

So we get that $LD+I=\beta{\mathbf{j}}^{T}$. $\blacksquare$

Remark 3.4

When $\alpha_{i}=\beta_{i}$ for each $1\leqslant i\leqslant r$, the above theorem implies results in [9]. So all the graphs there can be dealt with by the above theorem. Besides, we give new applications of the above theorem to weighted cactoid digraphs in the following section.

Remark 3.5

In Theorem 3.3, for some index $1\leqslant i\leqslant r$, the case $\lambda_{i}$ is not invertible may happen while $\lambda$ is invertible. This means the matrix $D$ is invertible doesn’t depend on whether each $D_{i}$ is invertible or not, but on the sum $\sum\limits_{i=1}^{r}\lambda_{i}$ is invertible.

4 The weighted cactoid digraph

An edge with a direction is called an arc. A directed path is a sequence of arcs with the same direction which is from each vertex to its successor in the sequence. A directed cycle is a directed path with its starting vertex and ending vertex coincide with each other. Each path (or cycle) in an undirected graph corresponds to two directed paths (or directed cycles) with opposite directions. Note that a directed cycle can have length two, but the minimum length of a cycle is three. A cactoid digraph is a strongly connected directed graph whose blocks are directed cycles. The inverse of the distance matrix of a cactoid digraph was studied in [6]. Here we study the weighted cactoid digraph which is a strongly connected directed graph with each of its block a weighted directed cycle. The weighted directed cycle ${\sf w}C_{n}$ is defined below.

Let $\mathbb{Z}$ be the ring of integers. Let $n\geqslant 2$, and let $\mathbb{Z}_{n}$ be the ring of integers modulo $n$. Let $\phi$ be the natural map from $\mathbb{Z}$ to $\mathbb{Z}_{n}$, and let

$X=\{0,1,\ldots,n-1\}\subseteq\mathbb{Z}$.

Then $\mathbb{Z}_{n}=\phi(X)=\{\phi(0),\phi(1),\ldots,\phi(n-1)\}$. Let ${\sf d}C_{n}$ be a directed cycle with vertex set $V({\sf d}C_{n})=\mathbb{Z}_{n}$ and arc set $E({\sf d}C_{n})=\{(\phi(i),\phi(i+1))\mid i\in X\}$. So for any $i\in X$ and $k\in\mathbb{Z}$, the vertex $\phi(i+kn)$ coincides with $\phi(i)$. Then the distance matrix of ${\sf d}C_{n}$ is

[TABLE]

where $P=(p_{\phi(i),\phi(j)})_{i,j\in X}$ is the cyclic permutation matrix whose nonzero entries are $p_{\phi(i),\phi(i+1)}=1$ ($i\in X$). Notice that the distance satisfying

[TABLE]

Let $\mathbb{R}$ be a commutative ring with identity. We use $\frac{1}{r}$ to denote the multiplicative inverse of an invertible element $r\in\mathbb{R}$. The weighted directed cycle ${\sf w}C_{n}$ is obtained from ${\sf d}C_{n}$ by giving each arc $(\phi(i),\phi(i+1))$ a weight $w_{i}\in\mathbb{R}$, where $i\in X$. Then the distance matrix of ${\sf w}C_{n}$ is ${\sf D}({\sf w}C_{n})=(d_{\phi(i),\phi(j)})_{i,j\in X}$, where

[TABLE]

Let

[TABLE]

We call $w$ and $w^{(2)}$ the first weight and the second weight of ${\sf w}C_{n}$, respectively.

Let $A=(a_{ij})$ be an $n\times n$ matrix. We denote its determinant by $\det(A)$ and $A(i|j)$ the submatrix obtained from $A$ by deleting the $i$-th row and $j$-th column. The cofactor of the entry $a_{ij}$ is $(-1)^{i+j}\det(A(i|j))$. The cofactor ${\rm cof}(A)$ of $A$ is defined as the sum of cofactors of the entries of $A$, i.e. ${\rm cof}(A)=\sum\limits_{i,j=1}^{n}(-1)^{i+j}\det(A(i|j))$. We denote the adjoint matrix of $A$ by ${\rm adj}(A)=(b_{ij})_{n\times n}$, i.e. $b_{ji}=(-1)^{i+j}\det(A(i|j))$. By the definitions, we have ${\rm cof}(A)={\mathbf{j}}^{T}{\rm adj}(A){\mathbf{j}}$. If $A$ is invertible, then $\displaystyle A^{-1}=\frac{{\rm adj}(A)}{\det(A)}$, and so we have

[TABLE]

Lemma 4.1

We use the above notation, and let $W={\sf D}({\sf w}C_{n})$. Suppose the first weight $w$ is invertible, and let

[TABLE]

Then

$\alpha^{T}{\mathbf{j}}=1$, $L{\mathbf{j}}=\mathbf{0}$, $\alpha^{T}W=\lambda{\mathbf{j}}^{T}$ and $LW+I=\beta{\mathbf{j}}^{T}$; and 2. 2.

${\mathbf{j}}^{T}\beta=1$, ${\mathbf{j}}^{T}L=\mathbf{0}$, $W\beta=\lambda{\mathbf{j}}$ and $WL+I={\mathbf{j}}\alpha^{T}$.

• Proof.  

The proofs of the two results are similar, so we only give the proof of the first one, i.e. we will show

$\alpha^{T}{\mathbf{j}}=1$, $L{\mathbf{j}}=\mathbf{0}$, $\alpha^{T}W=\lambda{\mathbf{j}}^{T}$ and $LW+I=\beta{\mathbf{j}}^{T}$.

The conditions $\alpha^{T}{\mathbf{j}}=1$ and $L{\mathbf{j}}=\mathbf{0}$ are obvious. Now we show $\alpha^{T}W=\lambda{\mathbf{j}}^{T}$. Let $W_{\phi(0)},W_{\phi(1)},\ldots,W_{\phi(n-1)}$ be all the column vectors of $W$, i.e.

$W=[W_{\phi(0)},W_{\phi(1)},\ldots,W_{\phi(n-1)}]$.

For $j\in X$, we have

[TABLE]

Hence we have $\alpha^{T}W=\lambda{\mathbf{j}}^{T}$.

At last we calculate $Y=LW+I=(y_{\phi(i),\phi(j)})_{i,j\in X}$. Let $i,j\in X$ with $i\neq j$. Then

[TABLE]

This means $wY=w\beta{\mathbf{j}}^{T}$. So we have $LW+I=\beta{\mathbf{j}}^{T}$. $\blacksquare$

Lemma 4.2

We use the above notation, and let $W={\sf D}({\sf w}C_{n})$. Suppose the first weight $w$ is invertible. Then the determinant of $W$ is

[TABLE]

• Proof.  

By elementary determinant evaluations, we get the following matrices $X_{1}=W$, $X_{2}$, X_{3}=\left(\begin{array}[]{cc}1&{\mathbf{j}}^{T}\\ \mathbf{0}&X_{2}\end{array}\right), $X_{4}$ and $X_{5}$. For $1\leqslant i\leqslant 5$, we use $R_{i}(j)$ to denote the $j$-th row of $X_{i}$ and $C_{i}(j)$ to denote the $j$-th column of $X_{i}$. The following are the elementary operations.

$X_{2}$ is obtained from $X_{1}$ by the following orerations:

[TABLE] 2. 2.

$X_{4}$ is obtained from $X_{3}$ by the following orerations:

[TABLE] 3. 3.

$X_{5}$ is obtained from $X_{4}$ by the following orerations:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In matrix $X_{5}$, the parameters

[TABLE]

By Lemma 4.1, we have $wX_{1}\beta=w^{(2)}{\mathbf{j}}$. Then for $1\leqslant j\leqslant n$, we get $wR_{1}(j)\beta=w^{(2)}$. So $-wb=w^{(2)}$. Hence, $\det(W)=\det(X_{2})=\det(X_{3})=\det(X_{4})=\det(X_{5})=-b(-w)^{n-1}=(-1)^{n-1}w^{n-2}w^{(2)}$. Note that, when the ring $\mathbb{R}$ is the real numbers, the formula of the determinant $\det(W)$ is still true even if the first weight $w$ is not invertible. $\blacksquare$

By Lemma 4.1, the distance matrix $W={\sf D}({\sf w}C_{n})$ of the weighted directed cycle ${\sf w}C_{n}$ is a left and right ${\sf LapExp}^{*}(\lambda,\alpha,\beta,L)$ matrix, where $\lambda$, $\alpha$, $\beta$ and $L$ are defined in Lemma 4.1. So the bag $B=(W,\lambda,\alpha,\beta,L)$ is a left and right ${\sf LapExp}^{*}$ $n$-bag. The bag $B$ is called the natural bag of the weighted directed cycle ${\sf w}C_{n}$. Suppose the first weight $w$ is invertible, then the determinant $\det(W)=(-1)^{n-1}w^{n-2}w^{(2)}$ by Lemma 4.2. If we assume the second weight $w^{(2)}$ is invertible additionally, then $\lambda$ is invertible and by Lemmas 2.1 and 2.2, $W$ is invertible,

$W^{-1}=-L+\frac{1}{\lambda}\beta\alpha^{T}=-L+\frac{w}{w^{(2)}}\beta\alpha^{T}$,

and $L$ is a Laplacian-like matrix. By Equation (20), we have

[TABLE]

As an application of Theorem 3.3, we get the ${\sf LapExp}^{*}$ property of the distance matrix of the weighted cactoid digraph.

Theorem 4.3

Let $G$ be a weighted cactoid digraph with structure parameters $(G^{n};G_{1}^{n_{1}},G_{2}^{n_{2}},\ldots,G_{r}^{n_{r}})$. For each $1\leqslant i\leqslant r$, suppose the first weight $w_{i}$ of the weighted directed cycle $G_{i}$ is invertible, and let $B_{i}=(W_{i},\lambda_{i},\alpha_{i},\beta_{i},L_{i})$ be the natural bag of $G_{i}$. Let $B=(W_{c},\lambda,\alpha,\beta,L)$ be the composition bag of natural bags $B_{1},B_{2},\ldots,B_{r}$, where $W_{c}$ is the distance matrix of $G$. Then $B$ is a left and right ${\sf LapExp}^{*}$ bag, and $L$ is a Laplacian-like matrix . Furthermore, if $\lambda=\sum\limits_{i=1}^{r}\lambda_{i}$ is invertible, then $W_{c}$ is invertible and $W_{c}^{-1}=-L+\frac{1}{\lambda}\beta\alpha^{T}$.

Graham, Hoffman and Hosoya [3] showed a very attractive theorem, expressing the determinant of the distance matrix of a distance well-defined graph explicitly as a function of its blocks. Applying their result to the weighted cactoid digraph in Theorem 4.3 and using Equations (21) and (22), we have

[TABLE]

If we take each edge as a directed cycle of length two with each arc weight one, then Equation (24) implies Graham and Pollak’s formula [5]

$\det({\sf D}(T_{n}))=(-1)^{n-1}2^{n-2}(n-1),$

and Theorem 4.3 implies the inverse of ${\sf D}(T_{n})$ [1, 4], where ${\sf D}(T_{n})$ is the distance matrix of the tree $T_{n}$ on $n$ vertices.

Since $w_{i}$ is invertible for each $1\leqslant i\leqslant r$, the distance matrix $W_{c}$ is invertible if and only if $\lambda$ is invertible by Equation (24). The above Theorem 4.3 implies part of the results in [2, 6, 8].

5 Classes of distance well-defined graphs

Recall that a graph is distance well-defined if it is a connected undirected graph, a strongly connected directed graph or a strongly connected mixed graph. So the distance matrix of a distance well-defined graph is well-defined. Here we define several classes of distance well-defined graphs.

A distance well-defined graph $G$ is called distance invertible, if the distance matrix ${\sf D}(G)$ is invertible. Let ${\sf DMI}$ be the set of distance invertible graphs. A distance well-defined graph $G$ is modified left (or right) Laplacian expressible, if the distance matrix ${\sf D}(G)$ is a left (or right) ${\sf LapExp}^{*}(\lambda,\alpha,\beta,L)$ matrix with $\lambda\neq 0$ and ${\mathbf{j}}^{T}\beta=1$ (or $\alpha^{T}{\mathbf{j}}=1$). A distance well-defined graph $G$ is modified Laplacian expressible, if either it is a modified left Laplacian expressible graph or it is a modified right Laplacian expressible graph. Let ${\sf LapExp}^{*}(\mathcal{L})$ be the set of modified left Laplacian expressible graphs. Let ${\sf LapExp}^{*}(\mathcal{R})$ be the set of modified right Laplacian expressible graphs. Let ${\sf LapExp}^{*}$ be the set of modified Laplacian expressible graphs. The relations between these classes of distance well-defined graphs are as follows,

${\sf LapExp}^{*}(\mathcal{L})\cup{\sf LapExp}^{*}(\mathcal{R})={\sf LapExp}^{*}\subseteq{\sf DMI}$.

Let $r\geqslant 2$ and $n_{i}\geqslant 2$ for each $1\leqslant i\leqslant r$. Let $G$ be the complete multipartite graph $K_{n_{1},n_{2},\ldots,n_{r}}$. If we suppose

$\#\{i\mid n_{i}=2,1\leqslant i\leqslant r\}\leqslant 1$,

then the distance matrix ${\sf D}(G)$ is invertible by Corollary 2.5 in [9]; furthermore, by the discussion on page 22 in [9], we know that the distance matrix ${\sf D}(G)$ is Laplacian expressible. So we have the following lemma.

Lemma 5.1

Let $r\geqslant 2$ and $n_{i}\geqslant 2$ for each $1\leqslant i\leqslant r$. Let $G$ be the complete multipartite graph $K_{n_{1},n_{2},\ldots,n_{r}}$. If $G\in{\sf DMI}$, then $G\in{\sf LapExp}^{*}$.

Hence the following question is interesting.

Question 5.2

Is there a distance well-defined graph $G\in{\sf DMI}\setminus{\sf LapExp}^{*}$?

Let ${\sf LapExp}_{*}={\sf LapExp}^{*}(\mathcal{L})\cap{\sf LapExp}^{*}(\mathcal{R})$. By examples in [7, 9] and Section 4, we get that ${\sf LapExp}_{*}\neq\emptyset$ and the following graphs are in ${\sf LapExp}_{*}$: the complete graph $K_{m}$ where $m\geqslant 2$, the odd cycle $C_{2k+1}$ where $k\geqslant 1$, the directed cycle $dC_{t}$ where $t\geqslant 2$, the complete multipartite graph $K_{n_{1},n_{2},\ldots,n_{r}}$ where $r\geqslant 2$, $n_{i}\geqslant 2$ for each $1\leqslant i\leqslant r$ and $\#\{i\mid n_{i}=2,1\leqslant i\leqslant r\}\leqslant 1$, the mixed complete graph ${\sf m}K_{s}$ where $s\geqslant 3$, and the weighted directed cycle ${\sf w}C_{n}$ where $n\geqslant 2$ and the weights satisfying $\lambda=\frac{w^{(2)}}{w}$ is invertible (see Lemma 4.1). By Remark 3.5, we have the following result.

Corollary 5.3

Let $G$ be a distance well-defined graph with structure parameters $(G^{n};G_{1}^{n_{1}},G_{2}^{n_{2}},\ldots,G_{r}^{n_{r}})$. Suppose for each $1\leqslant i\leqslant r$, the block $G_{i}$ is one of the following graphs: the complete graph $K_{m}$ where $m\geqslant 2$, the odd cycle $C_{2k+1}$ where $k\geqslant 1$, the directed cycle $dC_{t}$ where $t\geqslant 2$, the complete multipartite graph $K_{n_{1},n_{2},\ldots,n_{r}}$ where $r\geqslant 2$ and $n_{i}\geqslant 2$ for each $1\leqslant i\leqslant r$, the mixed complete graph ${\sf m}K_{s}$ where $s\geqslant 3$, and the weighted directed cycle ${\sf w}C_{n}$ where $n\geqslant 2$. Then the distance matrix of each $G_{i}$ is a left and right ${\sf LapExp}^{*}(\lambda_{i},\alpha_{i},\beta_{i},L_{i})$ matrix (may not invertible). We suppose $\lambda=\sum\limits_{i=1}^{r}\lambda_{i}$ is invertible. Then $G\in{\sf LapExp}_{*}$.

Corollary 5.3 implies many known results. We want to find more examples of graphs in ${\sf LapExp}_{*}$. So the following questions are interesting.

Question 5.4

*Is there a distance well-defined graph $G$ satisfying either $G\in{\sf LapExp}^{*}(\mathcal{R})\setminus{\sf LapExp}^{*}(\mathcal{L})$, or $G\in{\sf LapExp}^{*}(\mathcal{L})\setminus{\sf LapExp}^{*}(\mathcal{R})$? *

Question 5.5

How to characterize graphs in ${\sf LapExp}_{*}$, ${\sf LapExp}^{*}(\mathcal{L})$, ${\sf LapExp}^{*}(\mathcal{R})$, or ${\sf LapExp}^{*}$?

6 Acknowledgement

This work is supported by a project funded by China Postdoctoral Science Foundation under file number 2017M620491.