RECOGNIZING INDUCED SUBGRAPH AND TREE STRUCTURE FROM THE ADJACENCY SPECTRUM OF A GRAPH

Author

Emily Barranca, University of Rhode IslandFollow

Date of Award

2024

Degree Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics and Applied Mathematical Sciences

First Advisor

Michael D. Barrus, Jr.

Abstract

Let G = (V,E) be a simple, undirected graph. The adjacency spectrum of G is the multiset of eigenvalues of the adjacency matrix A(G) of G. We only consider the adjacency matrix so refer to the adjacency spectrum as the spectrum of the graph. In this paper, we discuss some connections between a graph's spectrum and its induced subgraphs. Additionally, we explore pairs of nonisomorphic graphs with the same spectrum, known as cospectral pairs.

Let G be a set of graphs. We define to be recognizable by spectrum if the spectrum of a graph H determines whether H contains any graph G ∈ G as an induced subgraph. If G is a singleton set containing only the graph G, we will say G is RS. We give some examples of RS sets, and we show K₁, K₂, K₃, 2K₁, and P₃ are RS. We conjecture these are the only RS graphs and show the conjecture holds for trees, unicyclic graphs, and some others.

Lastly, we examine the spectrum of diameter three trees called double stars, denoted P₂(a,b). We present many graphs cospectral to double stars with particular parameters a and b. Additionally, we show the double star P₂(1,n) for odd n is determined by its spectrum.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

Recommended Citation

Barranca, Emily, "RECOGNIZING INDUCED SUBGRAPH AND TREE STRUCTURE FROM THE ADJACENCY SPECTRUM OF A GRAPH" (2024). Open Access Dissertations. Paper 1678.
https://digitalcommons.uri.edu/oa_diss/1678