Laplacian spectral characterization of some double starlike trees

A tree is called double starlike if it has exactly two vertices of degree greater than two. Let $H(p,n,q)$ denote the double starlike tree obtained by attaching $p$ pendant vertices to one pendant vertex of the path $P_n$ and $q$ pendant vertices to the other pendant vertex of $P_n$. In this paper, we prove that $H(p,n,q)$ is determined by its Laplacian spectrum

Laplacian spectral characterization of roses

A rose graph is a graph consisting of cycles that all meet in one vertex. We show that except for two specific examples, these rose graphs are determined by the Laplacian spectrum, thus proving a conjecture posed by Lui and Huang [F.J. Liu and Q.X. Huang, Laplacian spectral characterization of 3-rose graphs, Linear Algebra Appl. 439 (2013), 2914--2920]. We also show that if two rose graphs have a so-called universal Laplacian matrix with the same spectrum, then they must be isomorphic. In memory of Horst Sachs (1927-2016), we show the specific case of the latter result for the adjacency matrix by using Sachs' theorem and a new result on the number of matchings in the disjoint union of paths

Spectral characterizations of propeller graphs

A propeller graph is obtained from an $\infty$-graph by attaching a path to the vertex of degree four, where an $\infty$-graph consists of two cycles with precisely one common vertex. In this paper, we prove that all propeller graphs are determined by their Laplacian spectra as well as their signless Laplacian spectra

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

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

Graphs whose second largest signless Laplacian eigenvalue does not exceed 2+sqrt(2)

For a graph G, let the signless Laplacian matrix Q(G) defined as Q(G)=D(G)+A(G), where A(G) and D(G) are, respectively, the adjacency matrix and the degree matrix of G. The Q-eigenvalues of G are the eigenvalues of Q(G). In this paper, we characterize the connected graphs whose second largest Q-eigenvalue κ2 does not exceed 2+2, obtain all the minimal forbidden subgraphs with respect to this property, and discover a large family of such graphs that are determined by their Q-spectrum. The connected graphs G such that κ2(G)=2+sqrt(2) are also detecte

On the Spectral Characterizations of Graphs

Several matrices can be associated to a graph, such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question “Which graphs are determined by their spectrum?” is still a difficult problem in spectral graph theory. Let $\mathcal{U}_p^{2q}$ be the set of graphs obtained from $C_p$ by attaching two pendant edges to each of $q (q \le p)$ vertices on $C_p$, whereas $\mathcal{V}_p^{2q}$ the subset of $\mathcal{U}_p^{2q}$ with odd $p$ and its $q$ vertices of degree 4 being nonadjacent to each other. In this paper, we show that each graph in $\mathcal{U}_p^{2q}$, $p$ even and its $q$ vertices of degree 4 being consecutive, is determined by its Laplacian spectrum. As well we show that if $G$ is a graph without isolated vertices and adjacency cospectral with the graph in $\mathcal{V}_p^{p−1} = \{ H \}$, then $G \cong H$

On the Spectral Characterizations of Graphs

Several matrices can be associated to a graph, such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question “Which graphs are determined by their spectrum?” is still a difficult problem in spectral graph theory. Let p2q${\cal U}_p^{2q}$ be the set of graphs obtained from Cp by attaching two pendant edges to each of q (q ⩽ p) vertices on Cp, whereas p2q${\cal V}_p^{2q}$ the subset of p2q${\cal U}_p^{2q}$ with odd p and its q vertices of degree 4 being nonadjacent to each other. In this paper, we show that each graph in p2q${\cal U}_p^{2q}$ , p even and its q vertices of degree 4 being consecutive, is determined by its Laplacian spectrum. As well we show that if G is a graph without isolated vertices and adjacency cospectral with the graph in pp−1={H}${\cal V}_p^{p - 1} = \{ H\}$ , then G ≅ H