‎Let $G$ be a non-abelian group and let $Gamma(G)$ be the non-commuting graph of $G$‎. ‎In this paper we define an equivalence relation $sim$ on the set of $V(Gamma(G))=Gsetminus Z(G)$ by taking $xsim y$ if and only if $N(x)=N(y)$‎, ‎where $ N(x)={uin G | x textrm{ and } u textrm{ are adjacent in }Gamma(G)}$ is the open neighborhood of $x$ in $Gamma(G)$‎. ‎We introduce a new graph determined ...

An independent set of vertices S of a graph dominates the graph efficiently if every vertex of the graph is either in S or has precisely one neighbour in S. In this paper we prove that a connected cubic vertex-transitive graph on a power of 2 vertices has a set that dominates it efficiently if and only if it is not isomorphic to a Möbius ladder. This is a preprint of an article accepted for pub...

Our paper has two goals: i) We propose the combinatorial approach to facilitate the calculation of the number of spanning trees for five new classes of graphs. ii) We use a new powerful operation (subdivision) to get larger graphs from a given graph. In particular, we derive the explicit formulas for the subdivision of ladder, fan, triangular snake, double triangular snake and the total graph o...

Let G be a (p, q) graph. Let f : V (G) → {1, 2, . . . , k} be a map. For each edge uv, assign the label gcd (f(u), f(v)). f is called k-prime cordial labeling of G if |vf (i) − vf (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labeled with x, ef (1) and ef (0) respectively denote the number of edges labeled with 1 and not labeled with 1....