The commuting graph of a finite non-abelian group G with center Z(G), denoted by ?c(G), is simple undirected whose vertex set G?Z(G), and two distinct vertices x y are adjacent if only xy=yx. Alwardi et al. (Bulletin, 2011, 36, 49-59) defined the common neighborhood matrix CN(G) energy Ecn(G) G. A called CN-hyperenergetic Ecn(G)>Ecn(Kn), where n=|V(G)| Kn denotes complete on n vertices. Two ...

A primitive digraph D is said to be well primitive if the local exponents of D are all equal. In this paper we consider well primitive digraphs of two special types: digraphs that contain loops, and symmetric digraphs with shortest odd cycle of length r. We show that the upper bound of the exponent of the well primitive digraph is n− 1 in both these classes of digraphs, and we characterize the ...

A kernel N of a digraph D is an independent set of vertices of D such that for every w ∈ V (D)− N there exists an arc from w to N . If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. D is called a critical kernel imperfect digraph when D has no kernel but every proper induced subdigraph of D has a kernel. If F is a set of arcs of D, a semikernel modulo F of...

It has been proved that sphericity testing for digraphs is an NP-complete problem. Here, we investigate sphericity of 3-connected single source digraphs. We provide a new combinatorial characterization of sphericity and give a linear time algorithm for sphericity testing. Our algorithm tests whether a 3-connected single source digraph with $n$ vertices is spherical in $O(n)$ time.

‎In this paper we defined the vertex removable cycle in respect of the following‎, ‎if $F$ is a class of graphs(digraphs)‎ ‎satisfying certain property‎, ‎$G in F $‎, ‎the cycle $C$ in $G$ is called vertex removable if $G-V(C)in in F $.‎ ‎The vertex removable cycles of eulerian graphs are studied‎. ‎We also characterize the edge removable cycles of regular‎ ‎graphs(digraphs).‎    

We consider 2-colored digraphs of the primitive ministrong digraphs having given exponents. In this paper we give bounds for 2-exponents of primitive extremal ministrong digraphs.

We construct a family of Cayley digraphs of degree d, diameter k and order kbd/2ck for any d ≥ 4 and k ≥ 3. We also present a collection of bipartite Cayley digraphs of order at least (k − 1)bd/2ck−1 for any degree d ≥ 4 and diameter k ≥ 4. For sufficiently large d and k, our digraphs are the largest known Cayley digraphs of degree d and diameter k, where k 6= d − 1 or d, and our bipartite digr...

Aharoni, R. and I. Ben-Arroyo Hartman, On Greene-Kleitman’s theorem for general digraphs, Discrete Mathematics 120 (1993) 13-24. Linial conjectured that Greene-Kleitman’s theorem can be extended to general digraphs. We prove a stronger conjecture of Berge for digraphs having k-optimal path partitions consisting of ‘long’ paths. The same method yields known results for acyclic digraphs, and exte...

A prominent problem in Graph Theory is to find extremal graphs or digraphs with restrictions in their diameter, degree and number of vertices. Here we obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and degree there is no other digraph with a smaller diameter. This new family is called modified cyclic digraphsMCK(d, ) and it is derived from the Kautz...