Let $S \subseteq R$. A matrix $M$ is said to be $S$-diagonalizable if there exists a matrix $P$ having entries from $S$ such that $M = P \Delta P^{−1}$ for some diagonal matrix $\Delta$. A graph $X$ is said to be $S$-diagonalizable if its Laplacian matrix $L := D − A$ is $S$-diagonalizable (Johnston and Plosker, 2023). Here, $D$ is the diagonal matrix of vertex degrees of $X$ and $A$ is the adjacency matrix of $X$. In this talk, we survey results on $S$-diagonalizable graphs, where $S = \{−1, 0, 1\}$ or $S = \{−1, 1\}$. In particular, we will focus on the case where $P$ is a Hadamard (that is, $P$ has entries from $\{-1,1\}$ and the columns of $P$ are pairwise orthogonal) or a weak Hadamard matrix (that is, $P$ has entries from $\{-1,0,1\}$ and the nonconsecutive columns of $P$ are pairwise orthogonal). Our main motivation for considering these graphs is its potential application to theory of continuous-time quantum walks, where knowledge about eigenvectors of certain matrices associated to graphs (more specifically, their orthogonal projection matrices) is important in determining which types of quantum state transfer occur in a graph. We also present open problems concerning $S$-diagonalizable graphs, if time allows. Some of the material presented in this talk is joint work with Darian McLaren and Sarah Plosker.

Let G be a graph. The maximum k-cut problem of a graph G is the maximum number of edges in a k-partite subgraph of G. The max k-cut problem is a fundamental NP-hard combinatorial optimization problem, and it is related to many applications such as data clustering, scheduling, power networks, and others. More recently, upper bounds to the max k-cut have been developed regarding the eigenvalues of a matrix related to graph G. In this talk, we present an extension of the graph parameter max k-cut to square matrices and prove a general sharp upper bound, which implies upper bounds on the max k-cut of a graph using the smallest signless Laplacian eigenvalue, the smallest adjacency eigenvalue, and the largest Laplacian eigenvalue of the graph. In addition, we show that our bounds are the best possible by constructing infinite families of extremal graphs for the obtained upper bounds.

This talk is about main eigenvalues of the signless Laplacian matrix of graphs. For a simple graph $G$ with vertices $v1, . . . ,vn$ , its signless Laplacian matrix $Q(G)=[qi j]$ is the square matrix of order n where $qii$ is equal to the degree of vertex $vi$ and, for $i \ne j, qij = 1$ if $vi$ and $vj$ are adjacent and 0, otherwise. An eigenvalue $\lambda$ is a main eigenvalue of $Q(G)$ (or a main $Q$-eigenvalue) if the eigenspace $E (\lambda)$ of $\lambda$ is non-orthogonal to the all ones vector $j$, that is, if $E (\lambda)$ contains some eigenvector for which the sum of its entries is non-zero. In a previous work, we gave an exact characterization of the threshold graphs having $k$ main $Q$-eigenvalues, for each $k \geq 2$. Here we give a step forward to generalize this characterization for quasi-threshold graphs. We may recall that threshold graphs form a subclass of cographs characterized as the class of $\{P4,C4,2K2\}$-free graphs, while quasi-threshold graphs constitute the subclass of $\{C4,P4\}$-free graphs . The main result of this work is a structural characterization of connected quasi-threshold graphs having $k \geq2$ main $Q$-eigenvalues. In addition, we determine precisely what are the connected quasi-threshold graphs having two main $Q$-eigenvalues. We also provide a full characterization of a subclass of quasi-threshold graphs: more precisely, for $k = 2,3, . . .$, we determine what are the generalized core-satellite graphs having $k$ main $Q$-eigenvalues. This is a joint work with Átila Jones and Vilmar Trevisan.

Zero forcing game starts with an initial set of blue and white vertices of a graph, and the goal is to color all vertices blue under some color change rule at minimum cost. The rule for the original zero forcing game is that if all of the neighbors of a blue vertex are blue except one, then the white neighbor is forced to be blue.
The zero forcing process applies this rule in discrete time-steps until no more changes are possible. The zero forcing number of a graph G is the minimum number Z(G) of vertices so that there exists an initial blue coloring with Z(G) vertices that is transformed to an all-blue coloring of G by the zero forcing process. The zero forcing was introduced to study the maximum multiplicity of eigenvalues of matrices associated with a graph. In this talk, some properties and applications of the zero forcing number will be discussed.