Unsolved Problems in Spectral Graph Theory

Spectral graph theory is a captivating area of graph theory that employs the eigenvalues and eigenvectors of matrices associated with graphs to study them. In this paper, we present a collection of $20$ topics in spectral graph theory, covering a range of open problems and conjectures. Our focus is primarily on the adjacency matrix of graphs, and for each topic, we provide a brief historical overview.Comment: v3, 30 pages, 1 figure, include comments from Clive Elphick, Xiaofeng Gu, William Linz, and Dragan Stevanovi\'c, respectively. Thanks! This paper will be published in Operations Research Transaction