Abstract
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) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.
Highlights
Let V(G) = {v1, v2, . . . vn} be the set of vertices in a connected graph G
We show that the complete graph Kn and the graph Kn − e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue among all connected graphs of order n
Assume that G is a connected graph of order n ≥ 3, and let v1 and v2 be the vertices with maximum total reciprocal distance vertex RHmax and the second maximum total reciprocal distance vertex RHmax, respectively
Summary
We, in this paper, study the second-largest signless Laplacian reciprocal distance eigenvalue of a connected graph. The signless Laplacian reciprocal distance eigenvalues of a connected graph G are linked to its connected spanning subgraph in the following lemma [22]. We discuss the relationship between the second-largest signless Laplacian reciprocal distance eigenvalues and the other graph parameters.
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