On regular graphs and coronas whose second largest eigenvalue does not exceed 1

We characterize all regular graphs whose second largest eigenvalue does not exceed 1. In the sequel, we determine all coronas, different from cones, with the same property. Some results and examples regarding unsolved cases are also given.     

Some graphs whose second largest eigenvalue does not exceed 2

We determine all trees whose second largest eigenvalue does not exceed $\sqrt{2}$. Next, we consider two classes of bipartite graphs, regular and semiregular, with small number of distinct eigenvalues. For all graphs considered we determine those whose second largest eigenvalue is equal to $\sqrt{2}$. Some additional results are also given.     

On conjectures involving second largest signless Laplacian eigenvalue of graphs

Let G=(V,E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the Laplacian matrix of G is L(G)=D(G)-A(G) and the signless Laplacian matrix of G is Q(G)=D(G)+A(G). In this paper we obtain a lower bound on the second largest signless Laplacian eigenvalue and an upper bound on the smallest signless Laplacian eigenvalue of G. In [5], Cvetkovi? et al. have given a series of 30 conjectures on Laplacian eigenvalues and signless Laplacian eigenvalues of G (see also [1]). Here we prove five conjectures.     

The largest eigenvalue of unicyclic graphs

Let G be a simple graph. Let λ₁(G) and μ₁(G) denote the largest eigenvalue of the adjacency matrix and the Laplacian matrix of G, respectively. Let Δ denote the largest vertex degree. If G has just one cycle, then

     

Graphs whose signless Laplacian spectral radius does not exceed the Hoffman limit value

For a graph matrix M, the Hoffman limit value H(M) is the limit (if it exists) of the largest eigenvalue (or, M-index, for short) of M(H_n), where the graph H_n is obtained by attaching a pendant edge to the cycle C_n-1 of length n-1. In spectral graph theory, M is usually either the adjacency matrix A or the Laplacian matrix L or the signless Laplacian matrix Q. The exact values of H(A) and H(L) were first determined by Hoffman and Guo, respectively. Since H_n is bipartite for odd n, we have H(Q)=H(L). All graphs whose A-index is not greater than H(A) were completely described in the literature. In the present paper, we determine all graphs whose Q-index does not exceed H(Q). The results obtained are determinant to describe all graphs whose L-index is not greater then H(L). This is done precisely in Wang et al. (in press) [21].     

Proof of conjecture involving the second largest signless Laplacian eigenvalue and the index of graphs

Let G = (V, E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the signless Laplacian matrix of G is Q(G) = D(G) + A(G). In [5], Cvetkovi? et al. have given the following conjecture involving the second largest signless Laplacian eigenvalue (q₂) and the index (λ₁) of graph G (see also Aouchiche and Hansen [1]):

     

On oriented graphs whose skew spectral radii do not exceed 2

Let S(G^σ)$S(G^{\sigma })$ be the skew adjacency matrix of the oriented graph G^σ$G^{\sigma }$ of order n   and _λ1,_λ2,…,_λn${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ be all eigenvalues of S(G^σ)$S(G^{\sigma })$. The skew spectral radius _ρs(G^σ)${\rho }_s(G^{\sigma })$ of G^σ$G^{\sigma }$ is defined as max{|_λ1|,|_λ2|,…,|_λn|}$\max \{|{\lambda }_1|,|{\lambda }_2|,\dots ,|{\lambda }_n|\}$. In this paper, we investigate oriented graphs whose skew spectral radii do not exceed 2.     

Some graphs with small second eigenvalue

For any prime,p, we construct a Cayley graph on the group,G, of affine linear transformations ofℤ/pℤ of degree 2(p−1) and second eigenvalue with the following special property: the adjacency matrix of the graph is supported on the “blocks” associated to the trivial representation and the irreducible representation of sizep−1. SinceG is of orderp(p−1), the correspondingt-uniform Cayley hypergraph has essentially optimal second eigenvalue for this degree and size of the graph (see [2] for definitions). En route we give, for any integerk>1, a simple Cayley graph onp ^k nodes of degreep of second eigenvalue . The author wishes to acknowledge the National Science Foundation for supporting this research in part under Grant CCR-8858788, and the Office of Naval Research under Grant N00014-87-K-0467.     

On the nullity of tricyclic graphs

The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. It is known that η(G)?n-2 if G is a simple graph on n vertices and G is not isomorphic to nK₁. The extremal graphs attaining the upper bound n-2 and the second upper bound n-3 have been obtained. In this paper, the graphs with nullity n-4 are characterized. Furthermore the tricyclic graphs with maximum nullity are discussed.     

On the index of tricyclic graphs with perfect matchings

Let T(2k) be the set of all tricyclic graphs on 2k(k?2) vertices with perfect matchings. In this paper, we discuss some properties of the connected graphs with perfect matchings, and then determine graphs with the largest index in T(2k).     

On tricyclic graphs of a given diameter with minimal energy

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let G(n,d) be the class of tricyclic graphs G on n vertices with diameter d and containing no vertex disjoint odd cycles C_p,C_q of lengths p and q with p+q≡2(mod4). In this paper, we characterize the graphs with minimal energy in G(n,d).     

On the third largest Laplacian eigenvalue of a graph

In this article, a sharp lower bound for the third largest Laplacian eigenvalue of a graph is given in terms of the third largest degree of the graph.     

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.     

On graphs whose second largest eigenvalue equals 1 – the star complement technique

The star complement technique is a spectral tool recently developed for constructing some bigger graphs from their smaller parts, called star complements. Here we first identify among trees and complete graphs those graphs which can be star complements for 1 as the second largest eigenvalue. Using the graphs just obtained, we next search for their maximal extensions, either by theoretical means, or by computer aided search.