Principal minor version of Matrix-Tree theorem for mixed graphs

Abstract

In Yu, et al. (2017), an analytical expression of the determinant of the Hermitian (quasi-)Laplacian matrix of mixed graphs has been proven. In this paper, we are going to extend those results and derive an analytical expression for the principal minors of the Hermitian (quasi-)Laplacian matrix, which is the principal minor version of the Matrix-Tree theorem.

Introduction

Let $G=\left(V\right(G),E(G\left)\right)$ be a simple undirected of Mechatronics and Biomedical Computer Science graph with vertex set V(G) and edge set E(G). A mixed graph M is obtained from an undirected graph G by orienting a subset of its edges. We call G the underlying graph of M, denoted by M_u. The vertex set of M is denoted by V(M), which equals V(G). The edge set E(M) is the union of the set of undirected edges E₀(M) and the set of directed edges (arcs) E₁(M). We distinguish an undirected edge by x↔y, while the directed edge (arcs) is to be x → y, if the orientation is from x to y. If we do not consider any direction, we just write xy is an edge of M.

Similar to the adjacency matrix of undirected graphs, Liu and Li [6] and Guo and Mohar [4] independently introduced a definition for the adjacency matrix of a mixed graphs as follows. The Hermitian adjacency matrix of a mixed graph M of order n is an n × n matrix $H\left(M\right)=\left(h_{kl}\right),$ where $h_{kl}=-h_{lk}=i$ ($i=\sqrt{-1}$), if there exists an orientation from v_k to v_l and $h_{kl}=h_{lk}=1,$ if there exists an edge between v_k and v_l without any orientation; otherwise $h_{kl}=0$.

The Hermitian Laplacian matrix L(M) of a mixed graph M has been introduced in [9]. It’s defined by $D\left(M\right)-H\left(M\right)$ where

$$D\left(M\right)=\text{diag}(d_1,d_2,\dots ,d_n)$$

is a diagonal matrix and d_i is the degree of the vertex v_i in the underlying graph M_u. The matrix $Q\left(M\right)=D\left(M\right)+H\left(M\right)$ is called the Hermitian quasi-Laplacian matrix of M, which has been introduced in [10]. Obviously these two matrices are Hermitian and all eigenvalues are real. Let $S\left(M\right)=\left(s_{ke}\right)$ be a n × m matrix indexed by the vertex and the edge of the mixed graph M, where s_ke is a complex number and $|s_{ke}|=1$ or 0. Moreover,

$$s_{ke}=\{\begin{array}{cc}-s_{le},\hfill & \text{if}{v}_k\leftrightarrow v_l\hfill \\ -i·s_{le},\hfill & \text{if}{v}_k\to v_l\hfill \\ i·s_{le},\hfill & \text{if}{v}_k\leftarrow v_l\hfill \\ 0,\hfill & \text{otherwise.}\hfill \end{array}$$

It is evident that S(M) is not unique. A matrix S(M) is referred to as incidence matrix of M. Moreover, $L\left(M\right)=S\left(M\right)S^{*}\left(M\right)$ and L(M) is positive semi-definite. A mixed graph is called Laplacian singular if its Hermitian Laplacian matrix is singular. Otherwise, it is Laplacian non-singular. Up to now, there has been only little research activity towards investigating the Hermitian spectra of a mixed graph (see [2], [4], [6], [8], [9], [10]).

A $i_1-i_k$-walk W in a mixed M is a sequence $W:v_{i_1}{v}_{i_2}\cdots v_{i_k}$ of vertices such that for $1\le s\le k-1$ we have $v_{i_s}\leftrightarrow v_{i_{s+1}}$ or $v_{i_s}\to v_{i_{s+1}}$ or $v_{i_s}\leftarrow v_{i_{s+1}}$. A $i_1-i_k$-walk W is called even (odd) if k is even (odd). The value of a mixed walk $W=v_1{v}_2{v}_3\cdots v_l$ is $h\left(W\right)=h_{12}{h}_{23}\cdots h_{(l-1)l}$. A mixed walk is positive or negative if $h\left(W\right)=1$ or $h\left(W\right)=-1,$ respectively. A mixed walk is called imaginary if $h\left(W\right)=\pm i$. Note that for one direction the value of a mixed walk or a mixed cycle is α; for the reverse direction its value is $\overline{\alpha }$. Thus, if the value of a mixed cycle is 1 (resp. $-1$) for one direction, then its value is 1 (resp. $-1$) for the reverse direction. In such situations, we just call this mixed cycle a positive (negative) mixed cycle without mentioning any direction. A graph is positive (resp. negative) if each of it’s mixed cycles is positive (resp. negative). An induced subgraph of M is an induced subgraph of its underlying graph G with the same orientation. For a subgraph H of M, let $M-H$ be the subgraph obtained from M by deleting all vertices of H and all incident edges. For V₁⊆V(M), $M-V_1$ is the subgraph obtained from M by deleting all vertices in V₁ and all their incident edges.

Bapat [1] introduced a real Laplacian matrix on mixed graph and investigated the Matrix-Tree Theorem based on this matrix. Inspired by Bapat et al. [1], we here study the same question in terms of the Hermitian (quasi-)Laplacian matrix.