Independent Domination of Middle Graph and Line Graph–Laplacian Approach

The Middle graph of a graph G , denoted by M(G), is a graph whose vertex set is V(G)UE(G) , and two vertices are adjacent if they are adjacent edges of G or one is a vertex and other is an edge incident with it. The Line graph of G , written L(G), is the simple graph whose vertices are the edges of G, with ef Є E(L(G)) when e and f have a common end vertex in G An independent dominating set in a graph is a set that is both dominating and independent.The independent domination number of G is the minimum size of an independent dominating set.In this paper, we prove the result for connected graph G, iM(G) ≥ Rank of the Laplacian matrix of G – Q((number of internal vertice in Longest path of G)/2) and For connected graph of G,iL(G)) ≥ number of vertices in G – Rank of the Laplacian matrix Q ((number of internal vertices in longest path in G)/3) where Q is the quotient.