On integral variations for roots of the Laplacian matching polynomial of graphs

TL;DR

This paper investigates how adding edges affects the roots of the Laplacian matching polynomial of graphs, showing that integral root variations are generally impossible under certain conditions.

Contribution

It provides new theoretical results on the impossibility of integral root variations in the Laplacian matching polynomial when edges are added under specific constraints.

Findings

Integral variation in one root is impossible.

Integral variation in two roots is also impossible under certain conditions.

Provides conditions under which roots do not change by integer values.

Abstract

In this paper, we study the Laplacian matching polynomial of a graph and the effect of adding edges to a graph on the roots (called Laplacian matching roots) of this polynomial. In particular, we investigate the conditions under which the Laplacian matching roots change by integer values. We prove that the Laplacian matching root integral variation in one place is impossible and the Laplacian matching root integral variation in two places is also impossible under some constraints.