Universal Adjacency Matrices with Two Eigenvalues

AMS Mathematics Subject Classification: 05C50.Adjacency matrix;Universal adjacency matrix;Laplacian matrix;signless Laplacian;Graph spectra;Eigenvalues;Strongly regular graphs

Strongly walk-regular graphs

We study a generalization of strongly regular graphs. We call a graph strongly walk-regular if there is an $\ell >1$ such that the number of walks of length $\ell$ from a vertex to another vertex depends only on whether the two vertices are the same, adjacent, or not adjacent. We will show that a strongly walk-regular graph must be an empty graph, a complete graph, a strongly regular graph, a disjoint union of complete bipartite graphs of the same size and isolated vertices, or a regular graph with four eigenvalues. Graphs from the first three families in this list are indeed strongly $\ell$-walk-regular for all $\ell$, whereas the graphs from the fourth family are $\ell$-walk-regular for every odd $\ell$. The case of regular graphs with four eigenvalues is the most interesting (and complicated) one. Such graphs cannot be strongly $\ell$-walk-regular for even $\ell$. We will characterize the case that regular four-eigenvalue graphs are strongly $\ell$-walk-regular for every odd $\ell$, in terms of the eigenvalues. There are several examples of infinite families of such graphs. We will show that every other regular four-eigenvalue graph can be strongly $\ell$-walk-regular for at most one $\ell$. There are several examples of infinite families of such graphs that are strongly 3-walk-regular. It however remains open whether there are any graphs that are strongly $\ell$-walk-regular for only one particular $\ell$ different from 3

The spectral characterization of graphs of index less than 2 with no path as a component

AbstractA graph is said to be determined by the adjacency and Laplacian spectrum (or to be a DS graph, for short) if there is no other non-isomorphic graph with the same adjacency and Laplacian spectrum, respectively. It is known that connected graphs of index less than 2 are determined by their adjacency spectrum. In this paper, we focus on the problem of characterization of DS graphs of index less than 2. First, we give various infinite families of cospectral graphs with respect to the adjacency matrix. Subsequently, the results will be used to characterize all DS graphs (with respect to the adjacency matrix) of index less than 2 with no path as a component. Moreover, we show that most of these graphs are DS with respect to the Laplacian matrix

On a signless Laplacian spectral characterization of T-shape trees

AbstractLet M be an associated matrix of a graph G (the adjacency, Laplacian and signless Laplacian matrix). Two graphs are said to be cospectral with respect to M if they have the same M spectrum. A graph is said to be determined by M spectrum if there is no other non-isomorphic graph with the same spectrum with respect to M. It is shown that T-shape trees are determined by their Laplacian spectra. Moreover among them those are determined by their adjacency spectra are characterized. In this paper, we identify graphs which are cospectral to a given T-shape tree with respect to the signless Laplacian matrix. Subsequently, T-shape trees which are determined by their signless Laplacian spectra are identified

On a Laplacian spectral characterization of graphs of index less than 2

AbstractA graph is said to be determined by the adjacency (respectively, Laplacian) spectrum if there is no other non-isomorphic graph with the same adjacency (respectively, Laplacian) spectrum. The maximum eigenvalue of A(G) is called the index of G. The connected graphs with index less than 2 are known, and each is determined by its adjacency spectrum. In this paper, we show that graphs of index less than 2 are determined by their Laplacian spectrum

Universal adjacency matrices with two eigenvalues

AbstractConsider a graph Γ on n vertices with adjacency matrix A and degree sequence (d1,…,dn). A universal adjacency matrix of Γ is any matrix in Span {A,D,I,J} with a nonzero coefficient for A, where D=diag(d1,…,dn) and I and J are the n×n identity and all-ones matrix, respectively. Thus a universal adjacency matrix is a common generalization of the adjacency, the Laplacian, the signless Laplacian and the Seidel matrix. We investigate graphs for which some universal adjacency matrix has just two eigenvalues. The regular ones are strongly regular, complete or empty, but several other interesting classes occur

A note on the Ramsey number of stars — Complete graphs

AbstractBoza et al. [L. Boza, M. Cera, P. García-Vázquez, M.P. Revuelta, On the Ramsey numbers for stars versus complete graphs, European J. Combin. 31 (2010) 1680–1688] gave the exact value of the multicolor Ramsey number R(K1,q1,…,K1,qn,Kp1,…,Kpm) in terms of R(Kp1,…,Kpm). In this note, we give a short proof of this result

Graphs with three distinct eigenvalues and largest eigenvalue less than 8

AbstractIn this paper we consider graphs with three distinct eigenvalues and, we characterize those with the largest eigenvalue less than 8. We also prove a simple result which gives an upper bound on the number of vertices of graphs with a given number of distinct eigenvalues in terms of the largest eigenvalue