Related papers: On co-edge-regular graphs with 4 distinct eigenval…
Sharp bounds on the least eigenvalue of an arbitrary graph are presented. Necessary and sufficient (just sufficient) conditions for the lower (upper) bound to be attained are deduced using edge clique partitions. As an application, we prove…
The problem of characterizing graphs with a prescribed number of main eigenvalues is a long-standing problem in spectral graph theory. Although some constructions are known, only a few produce infinite families of simple connected graphs…
We consider the family of undirected Cayley graphs associated with odd cyclic groups, and study statistics for the eigenvalues in their spectra. Our results are motivated by analogies between arithmetic geometry and graph theory.
In this paper, we give infinitely many examples of (non-isomorphic) connected $k$-regular graphs with smallest eigenvalue in half open interval $[-1-\sqrt2, -2)$ and also infinitely many examples of (non-isomorphic) connected $k$-regular…
Motivated by work of Haythorpe, Thomassen and the author showed that there exists a positive constant $c$ such that there is an infinite family of 4-regular 4-connected graphs, each containing exactly $c$ hamiltonian cycles. We complement…
A certain signed adjacency matrix of the hypercube, which Hao Huang used last year to resolve the sensitivity conjecture, is closely related to the unique, 4-cycle free, 2-fold cover of the hypercube. We develop a framework in which this…
The simplex graph $S(G)$ of a graph $G$ is defined as the graph whose vertices are the cliques of $G$ (including the empty set), with two vertices being adjacent if, as cliques of $G$, they differ in exactly one vertex. Simplex graphs form…
We give a combinatorial characterization of graphs whose normalized Laplacian has three distinct eigenvalues. Strongly regular graphs and complete bipartite graphs are examples of such graphs, but we also construct more exotic families of…
We study oriented graphs whose Hermitian adjacency matrices of the second kind have few eigenvalues. We give a complete characterization of the oriented graphs with two distinct eigenvalues, showing that there are only four such graphs. We…
Let $\lambda\geq2$ be an integer. For strongly regular graphs with parameters $(v, k, a,c)$ and smallest eigenvalue $-\lambda$, Neumaier gave two bounds on $c$ by using algebraic property of strongly regular graphs. In this paper, we will…
We give a survey on graphs with fixed smallest eigenvalue, especially on graphs with large minimal valency and also on graphs with good structures. Our survey mainly consists of the following two parts: (i) Hoffman graphs, the basic theory…
We use the line digraph construction to associate an orthogonal matrix with each graph. From this orthogonal matrix, we derive two further matrices. The spectrum of each of these three matrices is considered as a graph invariant. For the…
This paper addresses the challenge of spectral analysis and structural investigation for graphs that are not distance-regular, where computing the spectrum using standard methods based on equitable and orbit partitions can be complex. Our…
Two graphs are co-spectral if their respective adjacency matrices have the same multi-set of eigenvalues. A graph is said to be determined by its spectrum if all graphs that are co-spectral with it are isomorphic to it. We consider these…
Characterized are all simple undirected graphs $G$ such that any real symmetric matrix that has graph $G$ has no eigenvalues of multiplicity more than 2. All such graphs are partial 2-trees (and this follows from a result for rather general…
It has long been known that there exist finite connected tetravalent arc-transitive graphs with arbitrarily large vertex-stabilisers. However, beside a well known family of exceptional graphs, related to the lexicographic product of a cycle…
In [{Structural properties and decomposition of linear balanced matrices}, {\it Mathematical Programming}, 55:129--168, 1992], Conforti and Rao conjectured that every balanced bipartite graph contains an edge that is not the unique chord of…
We determine all graphs whose adjacency matrix has at most two eigenvalues (multiplicities included) different from $\pm 1$ and decide which of these graphs are determined by their spectrum. This includes the so-called friendship graphs,…
A locally irregular graph is a graph whose adjacent vertices have distinct degrees. It was conjectured that every connected graph is edge decomposable to $3$ locally irregular subgraphs, unless it belongs to a certain family of exceptions,…
We classify the distance-regular Cayley graphs with least eigenvalue $-2$ and diameter at most three. Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain…