Structural Characterizations and Algebraic Realizations of a Family of Regular Integral Graphs

Abstract: All the eigenvalues of an integral graphs are integers. Integral graphs are extremely rare. They form an asymptotically vanishing fraction $2<sup>{-Ω(n)}$ among all graphs on $n$ vertices. It makes the construction of a new family of integral graphs a challenging task. Also, most of the known infinite family of integral graphs rely on Cayley graphs over Abelian groups. In this article, we introduce a new family of integral graphs obtained from the groups. The construction of our graphs from groups is different from the construction of Cayley graphs. A spectral uniqueness theorem is established, which shows that each member of the infinite family is determined by its adjacency spectrum among all finite simple graphs. We also present recursive constructions that generates larger members of the family from smaller ones, providing a scalable class of integral graphs. Finally, we investigate algebraic realizations of these graphs as complements of Proper Prime Order Element Graphs of finite $2$-groups and obtain conditions characterizing such realizations. We also observe that the graphs obtained from different non-isomorphic groups have cospectral graphs.