The spread of generalized reciprocal distance matrix

TL;DR

This paper introduces the generalized reciprocal distance matrix for graphs, studies its eigenvalue spread, and establishes bounds and exact values for specific graph classes, extending previous related results.

Contribution

It defines the $RD_{\alpha}$-spread, derives bounds, and computes exact spreads for bipartite, clique-constrained, and double star graphs, generalizing prior work.

Findings

Established sharp bounds for $RD_{\alpha}$-spread.

Derived lower bounds for bipartite and clique-constrained graphs.

Calculated $RD_{\alpha}$-spread for double star graphs.

Abstract

The generalized reciprocal distance matrix $RD_{\alpha}(G)$ was defined as $RD_{\alpha}(G)=\alpha RT(G)+(1-\alpha)RD(G),\quad 0\leq \alpha \leq 1.$ Let $\lambda_{1}(RD_{\alpha}(G))\geq \lambda_{2}(RD_{\alpha}(G))\geq \cdots \geq \lambda_{n}(RD_{\alpha}(G))$ be the eigenvalues of $RD_{\alpha}$ matrix of graphs $G$. Then the $RD_{\alpha}$-spread of graph $G$ can be defined as $S_{RD_{\alpha}}(G)=\lambda_{1}(RD_{\alpha}(G))-\lambda_{n}(RD_{\alpha}(G))$. In this paper, we first obtain some sharp lower and upper bounds for the $RD_{\alpha}$-spread of graphs. Then we determine the lower bounds for the $RD_{\alpha}$-spread of bipartite graphs and graphs with given clique number. At last, we give the $RD_{\alpha}$-spread of double star graphs. Our results generalize the related results of the reciprocal distance matrix and reciprocal distance signless Laplacian matrix.