Spectral characterization of some graphs

The lexicographic product (or composition) of a cycle of length r with a totally disconnected graph on n vertices is shown to be a graph characterized by the eigenvalues of its adjacency matrix for all odd r.     

Spectral characterization of some weighted rooted graphs with cliques

The level of a vertex in a rooted graph is one more than its distance from the root vertex. A generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree. We characterize completely the eigenvalues of the Laplacian, signless Laplacian and adjacency matrices of a weighted rooted graph G obtained from a weighted generalized Bethe tree of k levels and weighted cliques in which

(1)

the edges connecting vertices at consecutive levels have the same weight,

(2)

each set of children, in one or more levels, defines a weighted clique, and

(3)

cliques at the same level are isomorphic.

These eigenvalues are the eigenvalues of symmetric tridiagonal matrices of order Moreover, we give results on the multiplicity of the eigenvalues, on the spectral radii and on the algebraic conectivity. Finally, we apply the results to the unweighted case and some particular graphs are studied.     

Spectral characterization of graphs whose second largest eigenvalue is less than 1

Graphs with second largest eigenvalue λ₂?1 are extensively studied, however, whether they are determined by their adjacency spectra or not is less considered. In this paper we completely characterize all the connected bipartite graphs with λ₂<1 that are determined by their adjacency spectra. In addition, we prove that all the connected non-bipartite graphs with girth no less than 4 and λ₂<1 are determined by their adjacency spectra.     

Spectral properties of Pascal graphs

We obtain spectral properties of the Pascal graphs by exploring its spectral graph invariants such as the algebraic connectivity, the first three largest Laplacian eigenvalues and the nullity. Some open problems pertaining to the Pascal graphs are given.     

Spectral characterizations of sandglass graphs

The sandglass graph is obtained by appending a triangle to each pendant vertex of a path. It is proved that sandglass graphs are determined by their adjacency spectra as well as their Laplacian spectra.     

Spectral radius and Hamiltonian graphs

In the paper, we will present two sufficient conditions for a bipartite graph to be Hamiltonian and a graph to be traceable, respectively.     

Spectral radius and Hamiltonicity of graphs

Let G be a graph of order n and μ(G) be the largest eigenvalue of its adjacency matrix. Let be the complement of G.Write K_n-1+v for the complete graph on n-1 vertices together with an isolated vertex, and K_n-1+e for the complete graph on n-1 vertices with a pendent edge.We show that:If μ(G)?n-2, then G contains a Hamiltonian path unless G=K_n-1+v; if strict inequality holds, then G contains a Hamiltonian cycle unless G=K_n-1+e.If , then G contains a Hamiltonian path unless G=K_n-1+v.If , then G contains a Hamiltonian cycle unless G=K_n-1+e.     

Spectral properties of complex unit gain graphs

A complex unit gain graph is a graph where each orientation of an edge is given a complex unit, which is the inverse of the complex unit assigned to the opposite orientation. We extend some fundamental concepts from spectral graph theory to complex unit gain graphs. We define the adjacency, incidence and Laplacian matrices, and study each of them. The main results of the paper are eigenvalue bounds for the adjacency and Laplacian matrices.