The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) the Harary (also called matrix) while diag(RH(G)) represents diagonal of total vertices. In present work, some upper and lower bounds second-largest eigenvalue graphs in terms various parameters are investigated. Besides, all attaining these new characterized. Addi...

The signless Laplacian spectral radius of a graph G, denoted by q(G), is the largest eigenvalue its matrix. In this paper, we investigate extremal for graphs without short cycles or long cycles. Let G(m,g) be family on m edges with girth g and H(m,c) circumference c. More precisely, obtain unique maximal q(G) in H(m,c), respectively.

Let Fk denote the k-fan consisting of k triangles which intersect in exactly one common vertex, and Sn,k complete split graph order n a clique on vertices an independent set remaining each vertex is adjacent to set. In this paper, it shown that unique attaining maximum signless Laplacian spectral radius among all graphs containing no Fk, provided k?2 n?3k2?k?2.