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

Abstract

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 vi. In 2022, generalized reciprocal distance matrix, which is defined by RDα(G)=αRT(G)+(1−α)RD(G),α∈[0,1], was introduced. In this paper, we give some bounds on the spectral radius of RDα(G) and characterize its extremal graph. In addition, we also give the generalized reciprocal distance spectral radius of line graph L(G).

Preface to the Special Issue “Algebraic Structures and Graph Theory”

“…, 1) is unique, minimizing the distance of the Laplacian spectral radius among the n-vertex bipartite graphs, with all having a diameter greater than or equal to 4. A similar argument is posed by the authors of [18], who find upper and lower bounds on the spectral radius of the generalized reciprocal distance matrix of a connected graph with n vertices. The main goal of Solovyev's paper [19] is to establish a counting formula for a 2-dimensional lattice path model with filter restrictions in the presence of long steps.…”