Related papers: Biregular graphs with three eigenvalues
An automorphism of a graph is called quasi-semiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malni\v{c}, Mart\'{i}nez and…
We complete the determination of the signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to $\pm 1$. The unsigned graphs and the disconnected, the bipartite and the complete signed graphs with this…
We initiate a systematic study of eigenvectors of random graphs. Whereas much is known about eigenvalues of graphs and how they reflect properties of the underlying graph, relatively little is known about the corresponding eigenvectors. Our…
We deal with connected $k$-regular multigraphs of order $n$ that has only three distinct eigenvalues. In this paper, we study the largest possible number of vertices of such a graph for given $k$. For $k=2,3,7$, the Moore graphs are…
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…
In this paper, we classify non-geometric distance-regular graphs of diameter at least $3$ with smallest eigenvalue at least $-3$. This is progress towards what is hoped to be an eventual complete classification of distance-regular graphs…
We introduce a new notation for representing labeled regular bipartite graphs of arbitrary degree. Several enumeration problems for labeled and unlabeled regular bipartite graphs have been introduced. A general algorithm for enumerating all…
In this paper we unify several existing regularity conditions for graphs, including strong regularity, $k$-isoregularity, and the $t$-vertex condition. We develop an algebraic composition/decomposition theory of regularity conditions. Using…
In this paper, we show that for given positive integer C, there are only finitely many distance-regular graphs with valency k at least three, diameter D at least six and k2/k<=C. This extends a conjecture of Bannai and Ito.
We characterize the graphs with loops whose degree sequences have no repeated values and find their adjacency spectrum. In the case of simple graphs, such graphs are called anti-regular graphs and are examples of threshold graphs. The…
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.
We study the minimum number of distinct eigenvalues over a collection of matrices associated with a graph. Lower bounds are derived based on the existence or non-existence of certain cycle(s) in a graph. A key result proves that every…
We prove that every finite arc-transitive graph of valency twice a prime admits a nontrivial semiregular automorphism, that is, a non-identity automorphism whose cycles all have the same length. This is a special case of the Polycirculant…
Let $F$ and $G$ be simple finite oriented graphs (without symmetric arcs). A graph $G$ is called $F$-irregular if any two distinct vertices in $G$ belong to a different number of subgraphs of $G$ isomorphic to $F$. In this paper, we…
For signed graphs we provide a cubic polynomial upper bound on the multiplicity of its eigenvalues. We show that this bound is sharp by providing examples of signed graphs in which it is attained. We also discuss particular cases in which…
For a given graph $G$, we aim to determine the possible realizable spectra for a generalized (or sometimes referred to as a weighted) Laplacian matrix associated with $G$. This new specialized inverse eigenvalue problem is considered for…
In this paper, we study the non-bipartite distance-regular graphs with valency k and having a smallest eigenvalue at most -k/2.
We present sharp inequalities relating the number of vertices, edges, and triangles of a graph to the smallest eigenvalue of its adjacency matrix and the largest eigenvalue of its Laplacian.
An eigenvalue of a graph $G$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. It is well known that a graph $G$ has exactly two main eigenvalues if and only if there exists a unique pair of…
A graph is said to be $k$-{\em isoregular} if any two vertex subsets of cardinality at most $k$, that induce subgraphs of the same isomorphism type, have the same number of neighbors. It is shown that no $3$-isoregular bicirculant (and more…