Characterizing Graphs as Algebraic Squares

TL;DR

This work formalizes squareness under the gluing algebra by introducing butterfly involutions and induced-subgraph criteria, then applies these tools to classify several graph families (cycles, paths, wheels, complete multipartite, circulants, and Johnson graphs) as square or non-square. It proves that squareness is preserved under key graph-operations when one factor is square (e.g., Cartesian, direct, strong, lexicographic product, and join), enabling the construction of infinite families of square graphs, while also providing counterexamples that show these closures are not converse. The results illuminate symmetry and substructure requirements for squareness and connect to broader themes in homomorphism densities and positive graphs. Open questions remain, notably a complete characterization for circulants and understanding squareness when combining two non-square graphs under common products. The framework offers practical criteria for constructing square graphs and guiding future investigations into the algebraic structure of graph products.

Abstract

Graphs that are squares under the gluing algebra arise in the study of homomorphism density inequalities such as Sidorenko's conjecture. Recent work has focused on these homomorphism density applications. This paper takes a new perspective and focuses on the graph properties of arbitrary square graphs, not only those relevant to homomorphism conjectures and theorems. We develop a set of necessary and/or sufficient conditions for a graph to be square. We apply these conditions to categorize several classical families of graphs as square or not. In addition, we create infinite families of square graphs by proving that joins and Cartesian, direct, strong, and lexicographic products of square graphs with arbitrary graphs are square.

Figures (10)

• Figure 1: A graph $B$ with a butterfly involution through the vertical line containing the two vertices of degree 6, colored red. Thus, $B$ is a square graph whose labeled vertices are the red vertices. Each gray vertex is paired with a black vertex. The line of symmetry of the graph is marked with a dashed line.
• Figure 2: Two examples of multiplication on partially labeled graphs in the gluing algebra are illustrated for graphs $G_1$ and $G_2$ in (a) and graphs $H_1$ and $H_2$ in (b).
• Figure 3: Two of the many butterfly involutions of the graph $B$ in Figure \ref{['fig:butterfly']}. Black and red vertices correspond to labeled vertices and gray vertices are unlabeled before the squaring operation. After the squaring operation, the "new" vertices are in yellow.
• Figure 4: Both components in $G-x$ are isomorphic to $P_4$, but $G$ is not a square.
• Figure 5: A butterfly involution of the even wheel $W_6$ is illustrated. Vertices $v_1$ and $v'_1$ are twinned, as are $v_2$ and $v'_2$.

Theorems & Definitions (56)

• Example 2.1
• Definition 2.3: AAP
• Definition 2.4
• Proposition 3.1
• proof
• Theorem 3.2
• proof
• Proposition 3.3
• proof
• Corollary 3.4