In this paper, we study structural properties of Toeplitz graphs. We characterize Kq-free graphs for an integer q≥3 and give equivalent conditions a graph Gn〈t1,t2,…,tk〉 with t1<⋯<tk n≥tk−1+tk being chordal Gn〈t1,t2〉 perfect. Then compute the edge clique cover number vertex graph. Finally, degree sequence (d1,d2,…,dn) n vertices show that is regular if only it circulant

A finite Frobenius group is a permutation group which is transitive but not regular such that only the identity element can fix two points. Such a group can be expressed as a semidirect product G = K o H, where K is a nilpotent normal subgroup. A first-kind G-Frobenius graph is a Cayley graph on K whose connection set is an H-orbit S on K that generates K, where H is of even order or S consists...

The action of a subgroup G of automorphisms of a graph X is said to be 2 -transitive if it is vertexand edgebut not arc-transitive. In this case the graph X is said to be (G, 2)-transitive. In particular, X is 1 2 -transitive if it is (Aut X, 1 2)-transitive. The 2 -transitive action of G on X induces an orientation of the edges of X which is preserved by G. Let X have valency 4. An even length...

A k-extended Skolem-type 5-tuple difference set of order t is a set of t 5-tuples {(di,1, di,2, di,3, di,4, di,5) | i = 1, 2, . . . , t} such that di,1+di,2+di,3+di,4+di,5 = 0 for 1 ≤ i ≤ t and {|di,j| | 1 ≤ i ≤ t, 1 ≤ j ≤ 5} = {1, 2, . . . , 5t+1}\{k}. In this talk, we will give necessary and sufficient conditions on t and k for the existence of a k-extended Skolem-type 5-tuple difference set ...

In this paper, we classify 2-closed (in Wielandt’s sense) permutation groups which contain a normal regular cyclic subgroup and prove that for each such group G, there exists a circulant Γ such that Aut(Γ) = G.

A 1-factorisation of a graph G is a decomposition of G into edge-disjoint 1-factors (perfect matchings), and a perfect 1-factorisation is a 1-factorisation in which the union of any two of the 1-factors is a Hamilton cycle. We consider the problem of the existence of perfect 1-factorisations of even order circulant graphs with small degree. In particular, we characterise the 3-regular circulant...

Knödel graphs and Fibonacci graphs are two classes of bipartite incident-graph of circulant digraphs. Both graphs have been extensively studied for the purpose of fast communications in networks, and they have deserved a lot of attention in this context. In this paper, we show that there exists an O(n log n)-time algorithm to recognize Knödel graphs, and that the same technique applies to Fibon...