On distance spectral radius of power hypertrees with given number of pendant paths of fixed length

TL;DR

This paper investigates the maximum and minimum distance spectral radius in specific power hypertrees with fixed pendant paths, identifying unique extremal structures under certain conditions.

Contribution

It characterizes the unique hypertrees that optimize the distance spectral radius among all r-th power hypertrees with given parameters.

Findings

Identifies the hypertrees that maximize the distance spectral radius.

Identifies the hypertrees that minimize the distance spectral radius.

Provides explicit conditions for extremal hypertrees based on parameters.

Abstract

The distance spectral radius of a connected hypergraph is the largest eigenvalue of the distance matrix of the hypergraph. A pendant path of length l with l greater than or equal to 1 in a hypergraph G at vertex v sub l plus 1 is a path consisting of vertices and edges v1 e1 v2 up to vl el v(l+1). The vertex v(l+1) has degree at least 2, vertex v1 has degree 1, and each vertex vi has degree 2 for i from 2 to l. Every vertex belonging to edge ei except vi and v(i+1) has degree 1 for all i from 1 to l. We find the unique hypertree that maximizes or minimizes the distance spectral radius among all r-th power hypertrees with m edges and k pendant paths of length l, where r is no less than 3, k and l are no less than 1, and the product of k and l is smaller than m.