Related papers: A Note on Eigenvalues of Cayley Graphs
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. In this paper, all connected tricyclic graphs with exactly two main eigenvalues are determined.
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. In this paper, all connected tricyclic graphs with exactly two main eigenvalues are determined.
A graph is said to be a bi-Cayley graph over a group H if it admits H as a group of automorphisms acting semiregularly on its vertices with two orbits. A non-abelian group is called an inner-abelian group if all of its proper subgroups are…
A mixed graph is called \emph{second kind hermitian integral}(or \emph{HS-integral}) if the eigenvalues of its Hermitian-adjacency matrix of second kind are integers. A mixed graph is called \emph{Eisenstein integral} if the eigenvalues of…
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…
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.
A nut graph is a simple graph for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry. It is known that infinitely many $d$-regular nut graphs exist for $3 \leq d \leq 12$…
A Cayley digraph on a group $G$ is called NNN if the Cayley digraph is normal and its automorphism group contains a non-normal regular subgroup isomorphic to $G$. A group is called NNND-group or NNN-group if there is an NNN Cayley digraph…
A graph is called odd (respectively, even) if every vertex has odd (respectively, even) degree. Gallai proved that every graph can be partitioned into two even induced subgraphs, or into an odd and an even induced subgraph. We refer to a…
New criteria for which Cayley graphs of cyclic groups of any order can be completely determined--up to isomorphism--by the eigenvalues of their adjacency matrices is presented. Secondly, a new construction for pairs of nonisomorphic Cayley…
A signless Laplacian eigenvalue of a graph $G$ is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, we first give the necessary and sufficient conditions for a…
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…
A signless Laplacian eigenvalue of a graph $G$ is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected bicyclic graphs with exactly two main…
A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group. In this paper we give conditions for when a Cayley graph on an abelian group can be represented as a…
A graph $\G$ with a group $H$ of automorphisms acting semiregularly on the vertices with two orbits is called a {\em bi-Cayley graph} over $H$. When $H$ is a normal subgroup of $\Aut(\G)$, we say that $\G$ is {\em normal} with respect to…
In this paper, generalized Cayley graphs are studied. It is proved that every generalized Cayley graph of order 2p is a Cayley graph, where p is a prime. Special attention is given to generalized Cayley graphs on Abelian groups. It is…
In a signed graph $G$, an induced subgraph is called a negative clique if it is a complete graph and all of its edges are negative. In this paper, we give the characteristic polynomials and the eigenvalues of some signed graphs having…
It is shown that a Cayley multigraph over a group $G$ with generating multiset $S$ is integral (i.e., all of its eigenvalues are integers) if $S$ lies in the integral cone over the boolean algebra generated by the normal subgroups of $G$.…
A group $G$ is complete group if it satisfies $Z(G)=e$ and $Aut(G)=Inn(G)$. In this paper, on the one hand, we study the basic properties of generalized Cayley graphs and characterize two classes isomorphic generalized generalized Cayley…
A map is a connected topological graph $\Gamma$ cellularly embedded in a surface. In this paper, applying Tutte's algebraic representation of map, new ideas for enumerating non-equivalent orientable or non-orientable maps of graph are…