A graph   is called integral if all eigenvalues of its adjacency matrix  are integers.  Given a subset $S$ of a finite group $G$, the bi-Cayley graph $BCay(G,S)$ is a graph with vertex set $Gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid sin S, xin G}$.  In this paper, we classify all finite groups admitting a connected cubic integral bi-Cayley graph.

Tree-width of graphs is a well studied notion, which plays an important role in structural graph theory and has many algorithmic applications. Various other graph invariants are known to be the same or within a constant factor of tree-width, for example, the bramble number or tangle number of a graph [4, 5], the branch-width [5], the linkedness [4], and the number of cops required to win the ro...

A k-uniform linear path of length l, denoted by P (k) l , is a family of k-sets {F1, . . . , Fl} such that |Fi ∩ Fi+1| = 1 for each i and Fi ∩ Fj = ∅ whenever |i − j| > 1. Given a k-uniform hypergraph H and a positive integer n, the k-uniform hypergraph Turán number of H , denoted by exk(n,H), is the maximum number of edges in a k-uniform hypergraph F on n vertices that does not contain H as a ...

By definition, Cayley graphs are vertex-transitive, and graphs underlying regular or orientably-regular maps (on surfaces) are arc-transitive. This paper addresses questions about how large the automorphism groups of such graphs can be. In particular, it is shown how to construct 3-valent Cayley graphs that are 5-arc-transitive (in answer to a question by Cai Heng Li), and Cayley graphs of vale...

For a finite undirected graph G = (V, E), let pu,v(t) denote the probability that a continuoustime random walk starting at vertex u is in v at time t. In this note we give an example of a Cayley graph G and two vertices u, v ∈ G for which the function ru,v(t) = pu,v(t) pu,u(t) t ≥ 0 is not monotonically non-decreasing. This answers a question asked by Peres in 2013.

It is known that a strongly regular semi-Cayley graph (with respect to a group G) corresponds to a triple of subsets (C, D, D') of G. Such a triple (C, D, D') is called a partial difference triple. First, we study the case when D U D' is contained in a proper normal subgroup of G. We basically determine all possible partial difference triples in this case. In fact, when \G\ f 8 nor 25, all part...

In this paper we introduce and study a type of Cayley graph – subnormal graph. We prove that 2-arc transitive is normal or cover complete bipartite $\mathbf{K}_{p^d,p^d}$ with $p$ prime. Then obtain generic method for constructing half-symmetric (namely edge but not arc transitive) graphs.

An automorphism of a graph is called quasi-semiregular if it fixes unique vertex the and its remaining cycles have same length. This kind symmetry graphs was first investigated by Kutnar, Malnič, Martínez Marušič in 2013, as generalization well-known problem regarding existence semiregular automorphisms vertex-transitive graphs. Symmetric valency three or four, admitting automorphism, been clas...