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Algebraic Structures on Graphs Joined by Edges

Daniel Pinzon, Daniel Pragel, Joshua Roberts

Abstract

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.

Algebraic Structures on Graphs Joined by Edges

Abstract

Let the join of two graphs be the union of two disjoint graphs connected by 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 -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 so that for any graph , the determinant of joined with and joined with is equal to the determinant of . Subsequently, we define a homomorphism from a quotient of graphs with the -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.
Paper Structure (12 sections, 16 theorems, 30 equations, 5 figures)

This paper contains 12 sections, 16 theorems, 30 equations, 5 figures.

Table of Contents

  1. Introduction
  2. An Equivalence Relation
  3. Equivalence Classes
  4. The Identity Class
  5. Algebraic Structure of the Join Operation
  6. The $[0]$ and $[n]$ classes
  7. Group Structure in the Semigroup
  8. Applications
  9. 1-join Example
  10. 2-join Example
  11. Chain of path graphs with different labeling
  12. Future Work

Key Result

Theorem 1.6

BP Let $G$ and $H$ be graphs of order $m$ and $n$ respectively where. Let $J$ be the set of vertices of $H$ that are joined to $G$. Then where the summation is over all allowable handle sets $B$.

Figures (5)

  • Figure 1: $j$-operation
  • Figure 2: Example of right identity for 1-join.
  • Figure 3: Determinant of $G \overset{2}{\asymp} H$
  • Figure 4: Directed edges from the copies of $P_2$ to $G$ are not necessary unless connectivity is desired.
  • Figure 5: Chain of $\widetilde{P}_4$

Theorems & Definitions (33)

  • Definition 1.1
  • Example 1.2
  • Definition 1.3
  • Definition 1.4
  • Example 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Definition 2.1
  • Theorem 2.2
  • ...and 23 more