The Characterizing Properties of (Signless) Laplacian Permanental Polynomials of Almost Complete Graphs

Tingzeng Wu,Tian Zhou,Barbara Martinucci

doi:10.1155/2021/9161508

Abstract

Let G be a graph with n vertices, and let L G and Q G denote the Laplacian matrix and signless Laplacian matrix, respectively. The Laplacian (respectively, signless Laplacian) permanental polynomial of G is defined as the permanent of the characteristic matrix of L G (respectively, Q G ). In this paper, we show that almost complete graphs are determined by their (signless) Laplacian permanental polynomials.

Highlights

We use G to denote a simple graph with vertex set V(G)􏼈v1, v2, . . . , vn􏼉 and edge set E(G) 􏼈e1, e2, . . . , em􏼉. e degree of a vertex v ∈ V(G) is denoted by d(v). e degree matrix of G, denoted by D(G), is the diagonal matrix whose (i, i)th entry is d(vi)
For a subgraph H of G, let G − E(H) denote the subgraph obtained from G by deleting the edges of H
Liu [19] showed that complete graph Kn and star Sn are determined by their Laplacian permanental polynomials

Summary

Introduction

􏼈v1, v2, . . . , vn􏼉 and edge set E(G) 􏼈e1, e2, . . . , em􏼉. e degree of a vertex v ∈ V(G) is denoted by d(v). e degree matrix of G, denoted by D(G), is the diagonal matrix whose (i, i)th entry is d(vi). E permanental polynomial of M, denoted by π(M, x), is defined to be the permanent of the characteristic matrix of M, i.e., π(M) π(M, x) per(xI − M),. Liu [19] showed that complete graph Kn and star Sn are determined by their (signless) Laplacian permanental polynomials. Let Gn denote the set of graphs each of which is obtained from Kn, by removing five or fewer edges. Our interest is to discuss which graph in Gn is detertmined by its (signless) Laplacian permanental polynomial. All graphs in Gn are determined by their (signless) Laplacian permanental polynomial.

Preliminaries

The Proof of Theorem 1