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 work of Badura and Rara, the determinant of the two joined graphs was decomposed to sums of determinants of these graphs with vertex deletions or directed graph handles. In this paper, we define a homomorphism from a quotient of graphs with the join operation to the monoid of integer matrices. We find the necessary and sufficient properties of a graph so that joining to any another graph will not change its determinant. We also demonstrate through examples that this decomposition allows us to more easily calculate determinants of chains of joined graphs. This paper begins the process of finding the algebraic structure of the monoid.