# The multiplicity of the Laplacian eigenvalue $2$ in some bicyclic graphs

## TL;DR

This paper examines the conditions under which the Laplacian eigenvalue 2 appears in graphs formed by connecting two unicyclic graphs with an edge, focusing on cases where only one or both graphs have 2 as an eigenvalue.

## Contribution

It extends previous work by analyzing the presence of the eigenvalue 2 in combined graphs when only one of the component graphs has this eigenvalue.

## Key findings

- Identifies structural conditions for eigenvalue 2 in combined graphs
- Provides Laplacian characteristic polynomial-based criteria
- Analyzes cases with one or both graphs having eigenvalue 2

## Abstract

The Laplacian matrix of a graph $G$ is denoted by $L(G)=D(G)-A(G)$, where $D(G)=diag(d(v_{1}),\ldots , d(v_{n}))$ is a diagonal matrix and $A(G)$ is the adjacency matrix of $G$. Let $G_1$ and $G_2$ be two graphs. A one-edge connection of two graphs $G_1$ and $G_2$ is a graph $G=G_1\odot_{uv} G_2$ with $V(G)=V(G_1)\cup V(G_2)$ and $E(G)= E(G_1)\cup E(G_2)\cup \{e=uv\}$, where $u\in V(G_1)$ and $v\in V(G_2)$. We investigate the multiplicity of the Laplacian eigenvalue $2$ of $G_1\odot_{uv} G_2$, while the unicyclic graphs $G_1$ and $G_2$ have $2$ among their Laplacian eigenvalues, by using their Laplacian characteristic polynomials. Some structural conditions ensuring the presence of the existence $2$ in the $G=G_1\odot_{uv} G_2$ where both $G_1$ and $G_2$ have $2$ as Laplacian eigenvalue, have been investigated, while, here we study the existence Laplacian eigenvalue $2$ in $G=G_1\odot_{uv} G_2$ where at most one of $G_1$ or $G_2$ has $2$ as Laplacian eigenvalue.