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 ∈ S : uv ∈ E(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 6= j and 0, otherwise. The graph G 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 to be bipartite and Hamiltonian. Also we obtain a bound for the number of edges of the graph G in terms of the structure of G. Further we obtain an upper bound for covering number (independence number) of G in terms of the covering number (independence number) of G.