On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

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 K-n and the graph K-n-e obtained from K-n by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

Automated summary

This study investigates upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters, characterizes all graphs attaining these new bounds, and concludes that complete graph K-n and graph K-n-e obtained from K-n by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue among all connected graphs with n vertices.