On trees with extremal extended spectral radius

TL;DR

This paper investigates the extremal values of the extended spectral radius in trees, identifying the path and star as the minimizer and maximizer respectively, and ranks the top five trees with the largest spectral radius.

Contribution

It characterizes extremal trees for the extended spectral radius, a novel matrix in graph theory, and determines the trees with the maximum spectral radius.

Findings

Paths minimize the extended spectral radius among trees.

Stars maximize the extended spectral radius among trees.

The first five trees with the largest extended spectral radius are identified.

Abstract

Let G be a simple connected graph with n vertices, and let d_i be the degree of the vertex v_i in G. The extended adjacency matrix of G is defined so that the ij-entry is 1/2(d_i/d_j+d_j/d_i) if the vertices v_i and v_j are adjacent in G, and 0 otherwise. This matrix was originally introduced for developing novel topological indices used in the QSPR/QSAR studies. In this paper, we consider extremal problems of the largest eigenvalue of the extended adjacency matrix (also known as the extended spectral radius) of trees. We show that among all trees of order n>= 5, the path Pn(resp., the star Sn) uniquely minimizes (resp., maximizes) the extended spectral radius. We also determine the first five trees with the maximal extended spectral radius.