Summary: For a graph \(G\) with vertex set \(V(G)=\{v_1, v_2, \dots, v_n\}\), let \(S\) be the covering set of \(G\) having the maximum degree over all the minimum covering sets of \(G\). Let \(N_S[v]=\{u\in S : uv \in E(G) \}\cup \{v\}\) 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 \(\left |N_S[v_i]\cap N_S[v_j] \right| \geq 1\), \(i\neq j\) 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 \(G\).