Graphs with Few Eigenvalues. An Interplay between Combinatorics and Algebra.

Abstract: Two standard matrix representations of a graph are the adjacency matrix and the Laplace matrix. The eigenvalues of these matrices are interesting parameters of the graph. Graphs with few eigenvalues in general have nice combinatorial properties and a rich structure. A well investigated family of such graphs comprises the strongly regular graphs (the regular graphs with three eigenvalues), and we may see other graphs with few eigenvalues as algebraic generalizations of such graphs. We study the (nonregular) graphs with three adjacency eigenvalues, graphs with three Laplace eigenvalues, and regular graphs with four eigenvalues. The last ones are also studied in relation with three-class association schemes. We also derive bounds on the diameter and on the size of special subsets in terms of the eigenvalues of the graph. Included are lists of feasible parameter sets of graphs with three Laplace eigenvalues, regular graphs with four eigenvalues, and three-class association schemes.

On graphs with just three distinct eigenvalues

Let G be a connected non-bipartite graph with exactly three distinct eigenvalues Rho, mu, lambda, where Rho >mu >lambda. In the case that G has just one non-main eigenvalue, we find necessary and sufficient spectral conditions on a vertex-deleted subgraph of G for G to be the cone over a strongly regular graph. Secondly, we determine the structure of G when just mu is non-main and the minimum degree of G is 1 + mu − lambda mu: such a graph is a cone over a strongly regular graph, or a graph derived from a symmetric 2-design, or a graph of one further type

Graphs with three and four distinct eigenvalues based on circulants

In this paper, we aim to address the open questions raised in various recent papers regarding characterization of circulant graphs with three or four distinct eigenvalues in their spectra. Our focus is on providing characterizations and constructing classes of graphs falling under this specific category. We present a characterization of circulant graphs with prime number order and unitary Cayley graphs with arbitrary order, both of which possess spectra displaying three or four distinct eigenvalues. Various constructions of circulant graphs with composite orders are provided whose spectra consist of four distinct eigenvalues. These constructions primarily utilize specific subgraphs of circulant graphs that already possess two or three eigenvalues in their spectra, employing graph operations like the tensor product, the union, and the complement. Finally, we characterize the iterated line graphs of unitary Cayley graphs whose spectra contain three or four distinct eigenvalues, and we show their non-circulant nature.Comment: 24 page

More on graphs with just three distinct eigenvalues

Let G be a connected non-regular non-bipartite graph whose adjacency matrix has spectrum ρ, µ(k) , λ(l) , where k, l ∈ IN and ρ > µ > λ. We show that if µ is non-main then δ(G) ≥ 1 + µ − λµ, with equality if and only if G is of one of three types, derived from a strongly regular graph, a symmetric design or a quasi-symmetric design (with appropriate parameters in each case)

Equiangular lines in Euclidean spaces

We obtain several new results contributing to the theory of real equiangular line systems. Among other things, we present a new general lower bound on the maximum number of equiangular lines in d dimensional Euclidean space; we describe the two-graphs on 12 vertices; and we investigate Seidel matrices with exactly three distinct eigenvalues. As a result, we improve on two long-standing upper bounds regarding the maximum number of equiangular lines in dimensions d=14, and d=16. Additionally, we prove the nonexistence of certain regular graphs with four eigenvalues, and correct some tables from the literature.Comment: 24 pages, to appear in JCTA. Corrected an entry in Table

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

On a conjecture about tricyclic graphs with maximal energy

For a given simple graph $G$, the energy of $G$, denoted by $\mathcal {E}(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix, which was defined by I. Gutman. The problem on determining the maximal energy tends to be complicated for a given class of graphs. There are many approaches on the maximal energy of trees, unicyclic graphs and bicyclic graphs, respectively. Let $P^{6,6,6}_n$ denote the graph with $n\geq 20$ vertices obtained from three copies of $C_6$ and a path $P_{n-18}$ by adding a single edge between each of two copies of $C_6$ to one endpoint of the path and a single edge from the third $C_6$ to the other endpoint of the $P_{n-18}$. Very recently, Aouchiche et al. [M. Aouchiche, G. Caporossi, P. Hansen, Open problems on graph eigenvalues studied with AutoGraphiX, {\it Europ. J. Comput. Optim.} {\bf 1}(2013), 181--199] put forward the following conjecture: Let $G$ be a tricyclic graphs on $n$ vertices with $n=20$ or $n\geq22$, then $\mathcal{E}(G)\leq \mathcal{E}(P_{n}^{6,6,6})$ with equality if and only if $G\cong P_{n}^{6,6,6}$. Let $G(n;a,b,k)$ denote the set of all connected bipartite tricyclic graphs on $n$ vertices with three vertex-disjoint cycles $C_{a}$, $C_{b}$ and $C_{k}$, where $n\geq 20$. In this paper, we try to prove that the conjecture is true for graphs in the class $G\in G(n;a,b,k)$, but as a consequence we can only show that this is true for most of the graphs in the class except for 9 families of such graphs.Comment: 32 pages, 12 figure