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.

Abstract : We call a Cayley graph Γ = Cay (G, S) normal for G, if the right regular representation R(G) of G is normal in the full automorphism group of Aut(Γ). In this paper, a classification of all non-normal Cayley graphs of finite abelian group with valency 6 was presented.  

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 ...

The concept of a coset-preserving skew-morphism is a generalization of the widely studied t-balanced skew-morphisms of regular Cayley maps which are in turn generalizations of group automorphisms. In case of abelian groups, all skew-morphisms of regular Cayley maps are roots of coset-preserving skew-morphisms, and therefore, classification of cosetpreserving skew-morphisms of finite abelian gro...

Up until the Bourgain-Gamburd 2005 breakthrough the only known ways to turn SLd(Fp) into an expander graph (i.e. to find a generating set of small size whose associated Cayley graph has a good spectral gap) was either through property (T ) (as in the Margulis construction) when d > 3 or through the Selberg property (and the dictionary between combinatorial expansion of the Cayley graphs and the...

In this paper, we introduce certain concepts, including neutrosophic hypergraph, line graph of neutrosophic hypergraph, dual neutrosophic hypergraph, tempered neutrosophic hypergraph and transversal neutrosophic hypergraph. We illustrate these concepts by several examples and investigate some of interesting properties. Key-words: Neutrosophic hypergraph, Line graph of neutrosophic hypergraph, D...

We consider a generalization of the notion of transversal to a finite hypergraph, so called weighted transversals. Given a non-negative weight vector assigned to each hyperedge of an input hypergraph A and a nonnegative threshold vector, we define a weighted transversal as a minimal vertex set which intersects all the hyperedges of A except for a subfamily of total weight not exceeding the give...

Skoviera, M. and J. ‘&Iii, Regular maps from Cayley graphs, Part i: Balanced Cayley maps, Discrete Mathematics IO9 (1992) 265-276. A Cayley map is a Cayley graph 2-cell embedded in some orientable surface so that the local rotations at every vertex are identical. Two types of Cayley maps are introduced: the balanced and antibalanced Cayley maps. In Part 1, conditions are given uuder which a bal...

The main purpose of this paper is to prove that if H is a 4-uniform hypergraph with n vertices and m edges, then the transversal number r(H) <2(m +n)/9. All standard terminology of hypergraphs is from [ 11. Suppose H = (V, E) is a k-uniform hypergraph with n vertices and m edges. Tuza [2] proposed the problem of finding an upper bound for the transversal number r(H), of the form t(H) < ck(n + m...

A linear algebraic group G over a field k is called a Cayley group if it admits a Cayley map, i.e., a G-equivariant birational isomorphism over k between the group variety G and its Lie algebra. A Cayley map can be thought of as a partial algebraic analogue of the exponential map. A prototypical example is the classical “Cayley transform” for the special orthogonal group SOn defined by Arthur C...