Quadratic starlike trees

TL;DR

This paper introduces quadratic graphs, focusing on quadratic starlike trees, fully characterizing them and providing their characteristic polynomials, thus advancing understanding of their spectral properties.

Contribution

The paper fully characterizes quadratic starlike trees and derives explicit expressions for their characteristic polynomials, expanding spectral graph theory.

Findings

Identified nine infinite families of quadratic starlike trees.

Provided explicit characteristic polynomial expressions.

Characterized all quadratic starlike trees including integral cases.

Abstract

In this paper, we introduce the notion of the quadratic graph, that is a graph whose eigenvalues are integral or quadratic algebraic integral, and determine nine infinite families of quadratic starlike trees, which are just all the quadratic starlike trees including integral starlike trees. Thus the quadratic starlike trees are completely characterized, and moreover, the display expressions for the characteristic polynomials of the quadratic starlike trees are also given.