A connection between the bipartite complements of line graphs and the line graphs with two positive eigenvalues

TL;DR

This paper explores the relationship between bipartite complements of line graphs and line graphs with specific eigenvalue properties, revealing connections between two classical graph families and providing new insights into their spectral characteristics.

Contribution

It establishes a link between bipartite complements of line graphs and line graphs with non-positive third eigenvalue, rederiving known results and highlighting structural connections.

Findings

Two graphs identified as both bipartite complements of line graphs and related to eigenvalue conditions

Reproves Borovianin's eigenvalue result using Courant-Weyl inequalities

Highlights structural features connecting classical graph families

Abstract

In 1974 Cvetkovi\'c and Simi\'c showed which graphs $G$ are the bipartite complements of line graphs. In 2002 Borovi\'canin showed which line graphs $L(H)$ have third largest eigenvalue $\lambda_3\leq0$. Our first observation is that two of the graphs Borovi\'canin found are the complements of two of the graphs found by Cvetkovi\'c and Simi\'c. Using the Courant-Weyl inequalities we show why this is and reprove the result of Borovi\'canin, highlighting some features of the graphs found by both.