Let Cay(H, S), S/H be an arbitrary Cayley graph over a finite group H. We shall say that two Cayley graphs Cay(H, S) and Cay(H, T) are Cayley isomorphic if there exists an automorphism . # Aut(H ) such that S=T. It is a trivial observation that two Cayley isomorphic Cayley graphs are isomorphic as graphs. The converse is not true: two Cayley graphs over the same group may be isomorphic as graph...

A complete characterization of locally primitive normal Cayley graphs of metacyclic groups is given. Namely, let Γ = Cay(G,S) be such a graph, where G ∼= Zm.Zn is a metacyclic group and m = p1 1 p r2 2 · · · p rt t such that p1 < p2 < · · · < pt. It is proved that G ∼= D2m is a dihedral group, and val(Γ ) = p is a prime such that p|(p1(p1 − 1), p2 − 1, . . . , pt − 1). Moreover, three types of ...

We review the concepts of hypertree decomposition and hypertree width from a graph theoretical perspective and report on a number of recent results related to these concepts. We also show – as a new result – that computing hypertree decompositions is fixed-parameter intractable. 1 Hypertree Decompositions: Definition and Basics This paper reports about the recently introduced concept of hypertr...

The original aim of this paper is to construct a graph associated to a vector space. By inspiration of the classical definition for the Cayley graph related to a group we define Cayley graph of a vector space. The vector space Cayley graph ${rm Cay(mathcal{V},S)}$ is a graph with the vertex set the whole vectors of the vector space $mathcal{V}$ and two vectors $v_1,v_2$ join by an edge whenever...

A Cayley (di)graph Cay(G,S) of a group G is called normal if the right regular representation in full automorphism Cay(G,S), and CI-(di)graph for every Cay(G,T),Cay(G,S)≅Cay(G,T) implies that there σ∈Aut(G) such Sσ=T. We call an NDCI-group or NCI-group all digraphs graphs are CI-digraphs CI-graphs, respectively. prove cyclic order n only 8∤n, either = 8 8∤n.

Normalized graph cut (NGC) has become a popular research topic due to its wide applications in a large variety of areas like machine learning and very large scale integration (VLSI) circuit design. Most of traditional NGC methods are based on pairwise relationships (similarities). However, in real-world applications relationships among the vertices (objects) may be more complex than pairwise, w...

For two normal edge-transitive Cayley graphs on groups H and K which have no common direct factor and $gcd(|H/H^prime|,|Z(K)|)=1=gcd(|K/K^prime|,|Z(H)|)$, we consider four standard products of them and it is proved that only tensor product of factors can be normal edge-transitive.

For a group T and a subset S of T , the bi-Cayley graph BCay(T, S) of T with respect to S is the bipartite graph with vertex set T×{0, 1} and edge set {{(g, 0), (sg, 1)} | g ∈ T, s ∈ S}. In this paper, we investigate cubic bi-Cayley graphs of finite nonabelian simple groups. We give several sufficient or necessary conditions for a bi-Cayley graph to be semisymmetric, and construct several infin...

In this paper, we determine all of connected normal edge-transitive Cayley graphs on non-abelian groups with order $4p^2$, where $p$ is a prime number.

Let be a connected Cayley graph of group G, then Γ is called normal if the right regular representation of G is a normal subgroup of , the full automorphism group of Γ. For the case where G is a finite nonabelian simple group and Γ is symmetric cubic Cayley graph, Caiheng Li and Shangjin Xu proved that Γ is normal with only two exceptions. Since then, the normality of nonsymmetric cubic Cayley ...