The existence of a connected 12-regular {K4, K2,2,2}-ultrahomogeneous graph G is established, (i.e. each isomorphism between two copies of K4 or K2,2,2 in G extends to an automorphism of G), with the 42 ordered lines of the Fano plane taken as vertices. This graph G can be expressed in a unique way both as the edge-disjoint union of 42 induced copies of K4 and as the edge-disjoint union of 21 i...

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set and edge set, but not arc set. It was shown by [Y.-Q. Feng, K.S. Wang, C.X. Zhou, Tetravalent halftransitive graphs of order 4p, European J. Combin. 28 (2007) 726–733] that all tetravalent half-arc-transitive graphs of order 4p for a prime p are non-Cayley and such graphs exist if andonly if p−1 is divi...

Transitive permutation groups having a non self paired suborbit of length are investigated via the corresponding orbital graphs If G is such a group and X is the orbital graph associated with a sub orbit of length of G which is not self paired then X has valency and admits a vertex and edge but not arc transitive action of G There is a natural balanced orientation of the edge set of X induced a...

Families of edge transitive algebraic graphs defined over finite commutative rings were used for the development of stream ciphers, public key cryptosystems and key exchange protocols. We present the results of the first implementation of a public key algorithm based on the family of algebraic graphs, which are not edge transitive. The absence of an edge transitive group of symmetries means tha...

This paper classifies all finite edge colored graphs with doubly transitive automorphism groups. This result generalizes the classification of doubly transitive balanced incomplete block designs with 1 and doubly transitive one-factorizations of complete graphs. It also provides a classification of all doubly transitive symmetric association schemes. The classification of finite simple groups i...

let g be a (p,q) graph. an injective map f : e(g) → {±1,±2,...,±q} is said to be an edge pair sum labeling if the induced vertex function f*: v (g) → z - {0} defi ned by f*(v) = σp∈ev f (e) is one-one where ev denotes the set of edges in g that are incident with a vertex v and f*(v (g)) is either of the form {±k1,±k2,...,±kp/2} or {±k1,±k2,...,±k(p-1)/2} u {±k(p+1)/2} according a...

Symmetric graph designs, or SGDs, were deened by Gronau et al. as a common generalisation of symmetric BIBDs and orthogonal double covers. This note gives a classiication of SGDs admitting a 2-transitive automorphism group. There are too many for a complete determination, but in some special cases the determination can be completed, such as those which admit a 3-transitive group, and those with...

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple (v, u, x, y) of vertices such that both (v, u, x) and (u, x, y) are paths of length two. The 3-arc graph of a graph G is defined to have vertices the arcs of G such that two arcs uv, xy are adjacent if and only if (v, u, x, y) is a 3-arc of G. In this paper we prove that any connected 3-arc graph is Hamiltonian, and all iterative 3-...

Computing the transitive closure in directed graphs is a fundamental graph problem. We consider the more restricted problem of computing the number of nodes reachable from every node and the size of the transitive closure. The fastest known transitive closure algorithms run in O(min{mn, n2.38}) time, where n is the number of nodes and m the number of edges in the graph. We present an O(m) time ...

In this research paper, we present a novel frame work for handling $m$-polar information by combining the theory of $m-$polar fuzzy  sets with graphs. We introduce certain types of edge regular $m-$polar fuzzy graphs and edge irregular $m-$polar fuzzy graphs. We describe some useful properties of edge regular, strongly edge irregular and strongly edge totally irregular $m-$polar fuzzy graphs. W...