On some covering graphs of a graph

For a graph G with vertex set V (G) = {v1; v2; : : : ; vn}, let S be the covering set of G having the maximum degree over all the minimum covering sets of G. Let NS[v] = {u є S : uv є E(G)g } {v} be the closed neighbourhood of the vertex v with respect to S: We define a square matrix AS(G) = (aij); by aij = 1; if │NS[vi] NS[vj ]│≥ 1; i ≠ j and 0, otherwise. The graph GS associated with the matrix AS(G) is called the maximum degree minimum covering graph (MDMC-graph) of the graph G. In this paper, we give conditions for the graph G^S to be bipartite and Hamiltonian. Also we obtain a bound for the number of edges of the graph G^S in terms of the structure of G. Further we obtain an upper bound for covering number (independence number) of GS in terms of the covering number (independence number) of G.