Two-place Laplacian matching root integral variations are impossible

TL;DR

This paper proves that two-place Laplacian matching root integral variations are impossible in all connected graphs, confirming a conjecture by Wang, Cui, and Cioab through structural analysis and new identities.

Contribution

It confirms the conjecture that no connected graph admits a two-place Laplacian matching root integral variation, using structural relations and novel power-sum identities.

Findings

Two-place Laplacian matching root integral variations do not exist in any connected graph.

The proof combines existing structural relations with new power-sum identities.

The conjecture by Wang, Cui, and Cioab is fully validated.

Abstract

Wang, Cui, and Cioab\u{a} introduced the Laplacian matching root integral variation of a graph and proved that it cannot occur in one place. They also showed that the two-place variation is impossible for connected graphs satisfying $g(G)/c(G)>7/6$, where $g(G)$ is the girth and $c(G)$ is the dimension of the cycle space, and conjectured that no connected graph admits such a two-place variation. In this paper, we confirm this conjecture. The proof combines a structural relation obtained in their paper with two new power-sum identities for Laplacian matching roots.