On some properties of the $\alpha$-spectral radius of the $k$-uniform hypergraph

TL;DR

This paper investigates how the $eta$-spectral radius of connected $k$-uniform hypergraphs varies with edge modifications, identifying extremal structures and second-largest values among certain hypergraph classes.

Contribution

It characterizes extremal hypergraphs for the $eta$-spectral radius under specific conditions and compares methods for identifying second-largest spectral radii.

Findings

Edge grafting affects the $eta$-spectral radius in predictable ways.

Extremal hypertrees for the $eta$-spectral radius are characterized.

The second-largest $eta$-spectral radius among $k$-uniform supertrees is identified.

Abstract

In this paper we show how the $\alpha$-spectral radius changes under the edge grafting operations on connected $k$-uniform hypergraphs. We characterize the extremal hypertree for $\alpha$-spectral radius among $k$-uniform non-caterpillar hypergraphs with given order, size and diameter. we also characerize the second largest $\alpha$-spectral radius among all $k$-uniform supertrees on $n$ vertices by two methods.