Characterizing Spectral Properties of Bridge

TL;DR

This paper analyzes the spectral properties of various bridge graphs, providing bounds on their Laplacian eigenvalues and laying groundwork for future studies on infinite bridge graphs.

Contribution

It introduces definitions and bounds for the second eigenvalues of Laplacians in different bridge graph types, advancing spectral graph theory.

Findings

Second eigenvalue of bridge graphs is between 0 and 2

Bounded eigenvalues for complete, star, and binary tree bridge graphs

Framework established for future analysis of infinite bridge graphs

Abstract

The Bridge graph is a special type of graph which are constructed by connecting identical connected graphs with path graphs. We discuss different types of bridge graphs $B_{n\times l}^{m\times k}$ in this paper. In particular, we discuss the following: complete-type bridge graphs, star-type bridge graphs, and full binary tree bridge graphs. We also bound the second eigenvalues of the graph Laplacian of these graphs using methods from Spectral Graph Theory. In general, we prove that for general bridge graphs, $B_{n\times l}^2$, the second eigenvalue of the graph Laplacian should be between $0$ and $2$, inclusive. In the end, we talk about future work on infinite bridge graphs. We created definitions and found the related theorems to support our future work about infinite bridge graphs.