Inertia Sets of Semicliqued Graphs

In this paper, we investigate inertia sets of simple connected undirected graphs. The main focus is on the shape of their corresponding inertia tables, in particular whether or not they are trapezoidal. This paper introduces a special family of graphs created from any given graph, $G$, coined semicliqued graphs and denoted $\widetilde{K}G$. We establish the minimum rank and inertia sets of some $\widetilde{K}G$ in relation to the original graph $G$. For special classes of graphs, $G$, it can be shown that the inertia set of $G$ is a subset of the inertia set of $\widetilde{K}G$. We provide the inertia sets for semicliqued cycles, paths, stars, complete graphs, and for a class of trees. In addition, we establish an inertia set bound for semicliqued complete bipartite graphs.