For any non-trivial abelian group A under addition a graph $G$ is said to be $A$-textit{magic}  if there exists a labeling $f:E(G) rightarrow A-{0}$ such that, the vertex labeling $f^+$  defined as $f^+(v) = sum f(uv)$ taken over all edges $uv$ incident at $v$ is a constant. An $A$-textit{magic} graph $G$ is said to be $Z_k$-magic graph if the group $A$ is $Z_k$  the group of integers modulo $k...

we find recursive formulae for the number of perfect matchings in a graph g by splitting g into subgraphs h and q. we use these formulas to count perfect matching of p hypercube qn. we also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is , where denotes the number of perfect matchings in g, is the graph constructed from by deleting edges with an en...

We present new edge splitting-off results maintaining all-pairs edge-connectivities of an undirected graph. We first give an alternate proof of Mader’s theorem, and use it to obtain a deterministic Õ(m + rmax · n2)-time complete edge splitting-off algorithm for unweighted graphs, where rmax denotes the maximum edge-connectivity requirement. This improves upon the best known algorithm by Gabow b...

We find recursive formulae for the number of perfect matchings in a graph G by splitting G into subgraphs H and Q. We use these formulas to count perfect matching of P hypercube Qn. We also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is , where denotes the number of perfect matchings in G, is the graph constructed from by deleting edges with an en...

The domination integrity of a connected graph G = (V (G), E(G)) is denoted as DI(G) and defined by DI(G) = min{|S| + m(G − S)}, where S is a dominating set and m(G − S) is the order of a maximum component of G− S. This paper discusses domination integrity of line splitting graph and central graph of some graphs.

In this paper, we determine the Randić energy of m-splitting graph, m-shadow graph and m-duplicate a given m being an arbitrary integer. Our results allow construction infinite sequence graphs having same energy. Further, some invariants like degree Kirchhoff index, Kemeny’s constant number spanning trees special graphs. From our results, indicate how to obtain infinitely many pairs equienerget...

Let H be a graph with a designated vertex s, where edges are weighted by nonnegative reals. Splitting edges e = {u, s} and e′ = {s, v} at s is an operation that reduces the weight of each of e and e′ by a real δ > 0 while increasing the weight of edge {u, v} by δ. It is known that all edges incident to s can be split off while preserving the edge-connectivity of H and that such a complete split...

A vertex set S is called a power dominating of graph G if every within the system monitored by following collection rules for grid monitoring. The domination number order minimal G. In this paper, we solve splitting and degree graph.