www.ejgta.org Electronic Journal of Graph Theory and Applications 4 (2) (2016), 132–147 On some covering graphs of a graph Shariefuddin Pirzada a , Hilal A. Ganie b , Merajuddin Siddique c a Department of Mathematics, University of Kashmir, Srinagar, Kashmir, India b Department of Mathematics, University of Kashmir, Srinagar, Kashmir, India c Department of Applied Mathematics, Aligarh Muslim University, Aligarh, India pirzadasd@kashmiruniversity.ac.in, hilahmad1119kt@gmail.com Abstract For a graph G with vertex set V (G)= {v 1 ,v 2 ,...,v n }, let S be the covering set of G having the maximum degree over all the minimum covering sets of G. Let N S [v]= {u ∈ S : uv ∈ E(G)}∪{v} be the closed neighbourhood of the vertex v with respect to S. We define a square matrix A S (G)=(a ij ), by a ij =1, if |N S [v i ] ∩ N S [v j ]|≥ 1,i = j and 0, otherwise. The graph G S associated with the matrix A S (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 G S in terms of the covering number (independence number) of G. Keywords: Covering graph, maximum degree, covering set, maximum degree minimum covering graph, covering number, independence number Mathematics Subject Classification : 05C15,05C50, 05C30 DOI:10.5614/ejgta.2016.4.2.2 1. Introduction Let G be finite, undirected, simple graph with n vertices and m edges having vertex set V (G)= {v 1 ,v 2 ,...,v n }. When the graph G is to be specified, the number of edges is denoted by m(G). A subset S of the vertex set V (G) is said to be covering set of G if every edge of G is incident to at least one vertex in S . A covering set with minimum cardinality among all covering sets Received: 9 June 2015, Revised 25 May 2016, Accepted: 4 September 2016. 132