On Complementary Distance Signless Laplacian Spectral Radius and Energy of Graphs

Let D be the diameter and d(G)(v(i), v(j)) be the distance between the vertices v(i) and v(j) of a connected graph G. The complementary distance matrix of a graph G is CD(G) = [cd(ij)] in which cd(ij) = 1 + D - d(G)(v(i), v(j)) if i not equal j and cd(ij) = 0 if i = j. The complementary transmission CTG(v) of a vertex v is defined as CTG(v) = Sigma(u is an element of V(G)) [1 + D d(G)(u, v)]. Let CT (G) = diag[CTG(v(1)) , CTG(v(2)), . . . , CTG (v(n))]. The complementary distance signless Laplacian matrix of G is CDL+(G) = CT(G) + CD(G). In this paper, we obtain the bounds for the largest eigenvalue of CDL+(G). Further we determine Nordhaus-Gaddum type results for the largest eigenvalue. We also establish some bounds for the complementary distance signless Laplacian energy.