The generalized reciprocal distance matrix of graphs

TL;DR

This paper introduces a new matrix $RD_{\alpha}(G)$ that interpolates between the reciprocal distance matrix and the reciprocal distance signless Laplacian, analyzing its eigenvalues, properties, bounds, and extremal graphs.

Contribution

It defines the generalized reciprocal distance matrix $RD_{\alpha}(G)$, explores its spectral properties, and identifies extremal graphs for various graph invariants.

Findings

Eigenvalues of $RD_{\alpha}(G)$ for special graphs are characterized.

Bounds for the spectral radius of $RD_{\alpha}(G)$ are established.

Extremal graphs with maximum spectral radius are identified for different graph parameters.

Abstract

Let $G$ be a simple undirected connected graph with the Harary matrix $RD(G)$, which is also called the reciprocal distance matrix of $G$. The reciprocal distance signless Laplacian matrix of $G$ is $RQ(G)=RT(G)+RD(G)$, where $RT(G)$ denotes the diagonal matrix of the vertex reciprocal transmissions of graph $G$. This paper intends to introduce a new matrix $RD_{\alpha}(G)=\alpha RT(G)+(1-\alpha)RD(G)$, $\alpha\in [0,1]$, to track the gradual change from $RD(G)$ to $RQ(G)$. First, we describe completely the eigenvalues of $RD_{\alpha}(G)$ of some special graphs. Then we obtain serval basic properties of $RD_{\alpha}(G)$ including inequalities that involve the spectral radii of the reciprocal distance matrix, reciprocal distance signless Laplacian matrix and $RD_{\alpha}$-matrix of $G$. We also provide some lower and upper bounds of the spectral radius of $RD_{\alpha}$-matrix. Finally, we depict the extremal graphs with maximal spectral radius of the $RD_{\alpha}$-matrix among all connected graphs of fixed order and precise vertex connectivity, edge connectivity, chromatic number and independence number, respectively.