Algebraic Combinatorics on Trace Monoids: Extending Number Theory to Walks on Graphs

Abstract

Partially commutative monoids provide a powerful tool to study graphs, viewing walks as words whose letters, the edges of the graph, obey a specific commutation rule. A particular class of traces emerges from this framework, the hikes, whose alphabet is the set of simple cycles on the graph. We show that hikes characterize undirected graphs uniquely, up to isomorphism, and satisfy remarkable algebraic properties such as the existence and uniqueness of a prime factorization. Because of this, the set of hikes partially ordered by divisibility hosts a plethora of relations in direct correspondence with those found in number theory. Some applications of these results are presented, including a permanantal extension to MacMahon's master theorem and a derivation of the Ihara zeta function.

A Hopf algebra for counting cycles

“…We briefly recall the definition and main properties of hikes. We refer to [12] for further details. Let P denote the set of simple cycles in G. Hikes are defined as the partially commutative monoid H with alphabet P and independence…”

“…The hike formalism is perfectly suited to describe the analytic properties of the graph G via its labeled adjacency matrix W, defined by W ij := ω ij if (i, j) ∈ E and W ij := 0 otherwise. In particular, the labeled adjacency matrix W preserves the partially commutative structure of the hikes provided that the edges ω ij are endowed with the commutation rule: ω ij ω i j = ω i j ω ij if i = i [12]. Thanks to this property, formal series on hikes can be represented as functions of this matrix and manipulated via its analytical transformations.…”

“…In our view both problems can be treated with the same tools rooted in the algebraic combinatorics of paths [12]. The literature of graph theory contains many different approaches to defining paths on graphs as algebraic objects.…”

“…Relaxing the connectedness condition, Cartier and Foata showed that hikes admit a simple description as words on the alphabet of graph edges and provide a powerful partially commutative framework for algebraic combinatorics on graphs. Recent developments [10,12] have shown that hikes are also words on the alphabet of simple cycles of the graph. In this section we recall a few essential results pertaining to hikes.…”

A centrality measure for cycles and subgraphs II

“…The dominant complex, here denoted Co1, comprises 30 proteins 6 and is found in both the MIPS database and in [17], where it is known as the mitochondrial small ribosomal large subunit. Interestingly, Co1 is identical with the third largest complex recovered by the MCL algorithm running on the same dataset [9], with the addition of the proteins ASF1 and MAM33, a nucleosome assembly factor and a protein of the mitochondrial matrix involved in oxidative phosphorylation, respectively.…”