Upper and Lower Bounds for the Spectral Radius of Generalized Reciprocal Distance Matrix of a Graph

For a connected graph G on n vertices, recall that the reciprocal distance signless Laplacian matrix of G is defined to be RQ(G) = RT (G) + RD (G), where RD (G) is the reciprocal distance matrix, RT (G) = diag(RT1, RT2, ..., RTn) and RTi is the reciprocal distance degree of vertex v(i). In 2022, generalized reciprocal distance matrix, which is defined by RD alpha(G) = alpha RT(G) + (1 - alpha)RD(G), alpha is an element of [0,1], was introduced. In this paper, we give some bounds on the spectral radius of RD alpha(G) and characterize its extremal graph. In addition, we also give the generalized reciprocal distance spectral radius of line graph L(G).

Automated summary

This study focuses on the generalized form of the reciprocal distance Laplacian matrix and its applications in graph theory. By investigating properties such as spectral radius and extremal graphs, it helps to provide a deeper understanding of the structure and characteristics of graphs.