Algebraic Structures on Graphs Joined by Edges

Let the join of two graphs be the union of two disjoint graphs connected by $j$ edges in a one-to-one manner. In previous work by Gyurov and Pinzon, which generalized the results of Badura and Rara, the determinant of the adjacency matrix of two $j$-joined graphs was decomposed to sums of determinants of these graphs with vertex deletions or directed graph handles. In this paper, we find the necessary and sufficient properties of a graph $G$ so that for any graph $H$, the determinant of $G$ joined with $H$ and $H$ joined with $G$ is equal to the determinant of $H$. Subsequently, we define a homomorphism from a quotient of graphs with the $j$-join operation to the monoid of integer matrices under multiplication. We demonstrate through examples that this homomorphism allows us to more easily calculate determinants of chains of joined graphs. This generalizes the work done on determinants of grids and cylinders done in various other works.