Graphs determined by their generalized characteristic polynomials

AbstractFor a given graph G with (0,1)-adjacency matrix AG, the generalized characteristic polynomial of G is defined to be ϕG=ϕG(λ,t)=det(λI-(AG-tDG)), where I is the identity matrix and DG is the diagonal degree matrix of G. In this paper, we are mainly concerned with the problem of characterizing a given graph G by its generalized characteristic polynomial ϕG. We show that graphs with the same generalized characteristic polynomials have the same degree sequence, based on which, a unified approach is proposed to show that some families of graphs are characterized by ϕG. We also provide a method for constructing graphs with the same generalized characteristic polynomial, by using GM-switching

Laplacian spectral characterization of some graph products

This paper studies the Laplacian spectral characterization of some graph products. We consider a class of connected graphs: $\mathscr{G}={G : |EG|\leq|VG|+1}$, and characterize all graphs $G\in\mathscr{G}$ such that the products $G\times K_m$ are $L$-DS graphs. The main result of this paper states that, if $G\in\mathscr{G}$, except for $C_{6}$ and $\Theta_{3,2,5}$, is $L$-DS graph, so is the product $G\times K_{m}$. In addition, the $L$-cospectral graphs with $C_{6}\times K_{m}$ and $\Theta_{3,2,5}\times K_{m}$ have been found.Comment: 19 pages, we showed that several types of graph product are determined by their Laplacian spectr

Which wheel graphs are determined by their Laplacian spectra?

The wheel graph, denoted by Wn+1, is the graph obtained from the circuit C-n with n vertices by adding a new vertex and joining it to every vertex of C-n. In this paper, the wheel graph Wn+1. except for W-7, is proved to be determined by its Laplacian spectrum, and a graph cospectral with the wheel graph W-7 is given. (C) 2009 Elsevier Ltd. All rights reserved

Computers and Mathematics with Applications Which wheel graphs are determined by their Laplacian spectra?

a b s t r a c t The wheel graph, denoted by W n+1 , is the graph obtained from the circuit C n with n vertices by adding a new vertex and joining it to every vertex of C n . In this paper, the wheel graph W n+1 , except for W 7 , is proved to be determined by its Laplacian spectrum, and a graph cospectral with the wheel graph W 7 is given