Related papers: Spectra and eigenspaces of non-normal Cayley graph…
We show that the star graph defined as the Cayley graph of ${\mathfrak S}_{n+1}$ generated by the star transpositions is an ${\mathfrak S}_n$-cover of the complete graph $K_{n+1}$, which is known to have fine spectral properties. In the…
Many natural and social systems develop complex networks, that are usually modelled as random graphs. The eigenvalue spectrum of these graphs provides information about their structural properties. While the semi-circle law is known to…
Let $G$ be a finite group, and $S$ be a subset of $G\setminus\{1\}$ such that $S=S^{-1}$. Suppose that $Cay(G,S)$ is the Cayley graph on $G$ with respect to the set $S$ which is the graph whose vertex set is $G$ and two vertices $a,b\in G$…
We expand the structural theory of \ca graphs that avoid specific cyclic coset patterns. We present several characterisations of tree-likeness for these structures and show a close connection to $\alpha$-acyclic hypergraphs. A focus lies on…
We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a…
The $(n,k)$-star graphs are an important class of interconnection networks that generalize star graphs, which are superior to hypercubes. In this paper, we continue the work begun by Cheng et al.~(Graphs and Combinatorics 2017) and complete…
In a prior work, the galaxies of the nonstandard enlargements of conventionally infinite graphs and also of transfinite graphs of the first rank of transfiniteness were defined, examined, and illustrated by some examples. In this work it is…
A discrete analog of quantum unique ergodicity was proved for Cayley graphs of quasirandom groups by Magee, Thomas and Zhao. They show that for large graphs there exist real orthonormal basis of eigenfunctions of the adjacency matrix such…
We show that the topological space of any infinite graph and its ends is normal. In particular, end spaces themselves are normal.
In this article we construct a series of new infinite families of strongly regular graphs with the same parameters as the point-graphs of non-singular quadrics in PG(n,2).
We give an introduction to the Cayley-Abels graph for a totally disconnected, locally compact (tdlc) group. It is a generalization of the Cayley graph. We illustrate that on the one hand, Cayley-Abels graphs are useful tools to extend…
This is a revision of the paper archived previously on August 22, 2002. It corrects a mistake in Sec. 8 concerning eccentricities of graphs. From any given sequence of finite or infinite graphs, a nonstandard graph is constructed. The…
A characterization is given of finite groups $H$ that have skew-morphisms of order coprime to the order $|H|$, and their skew-morphisms. A complete classification is then given of the automorphism groups and the underlying graphs of…
In this paper, we show that if G is strongly regular then the Gallai graph and the anti-Gallai graph of G are edge-regular. We also identify conditions under which the Gallai and anti-Gallai graphs are themselves strongly regular, as well…
The isomorphism problem of Cayley graphs has been well studied in the literature, such as characterizations of CI (DCI)-graphs and CI (DCI)-groups. In this paper, we generalize these to vertex-transitive graphs and establish parallel…
Let $G$ denote a finite abelian group with identity 1 and let $S$ denote an inverse-closed subset of $G \setminus {1}$, which generates $G$ and for which there exists $s \in S$, such that $\la S \setminus \{s,s^{-1}\} \ra \ne G$. In this…
The extremal eigenvalues including maximum eigenvalues and the minimum eigenvalues about outerplanar graphs are investigated in this paper. Some structural characterizations about the (edge) maximal bipartite outerplanar graphs are…
We study covering graphs of the Paley graph associated to a finite field of characteristic p in the case where the covering transformation group is cyclic of prime order distinct from p. When the field has q = p elements, we show that the…
A graph is said to be uniquely hamiltonian if it has a unique hamiltonian cycle. For a natural extension of this concept to infinite graphs, we find all uniquely hamiltonian vertex-transitive graphs with finitely many ends, and also discuss…
We give the characteristic polynomial of the distance or adjacency matrix of the strong power graph of a finite group, and compute its distance and adjacency spectrum.