Title: Graphs that Admit Orthogonal Matrices
| Speaker: | Shaun Fallat |
| Affiliation: | University of Regina |
| Location: | Please contact Sabrina Lato for Zoom link. |
Abstract: Given a simple graph $G=(\{1,\ldots, n},E), we consider the class $S(G)$ of real symmetric $n \times n$ matrices $A=[a_{ij}]$ such that for $i\neq j$, $a_{ij}\neq 0$ iff $ij \in E$. Under the umbrella of the inverse eigenvalue problem for graphs (IEPG), $q(G)$ - known as the minimum number of distinct eigenvalues of $G$ - has emerged as one of the most well-studied parameters of the IEPG. Naturally, characterizing graphs $G$ for which $q(G) \leq, =, \geq k$ is an important step for studying the IEPG. A tantalizing (and maddened) puzzle is to characterize the graphs $G$ with $q(G)=2$. Equivalently, such graphs are precisely the graphs for which $S(G)$ admits an orthogonal matrix. In this talk I intend to discuss the continuing saga of attempting to learn more about the graphs $G$ with $q(G)=2$...