1. What is the significance of trace monoids and hike monoids in research?
Trace monoids and hike monoids are significant in research as they provide a combinatorial approach to understanding and proving mathematical theorems. Trace monoids were introduced by Pierre Cartier and Dominique Foata in their quest for a purely combinatorial proof of MacMahon's master theorem. They are used to study the partially commutative structure induced by independence relations, represented by the dependency graph of the trace monoid. Hike monoids, on the other hand, are a specific instance of trace monoids where the numbers of incoming and outgoing edges at each vertex are equal. They have a simpler presentation as partially commutative monoids on the alphabet of directed graph simple cycles. Hike monoids are important in graph theory and have applications in various fields such as computer science, combinatorics, and algebraic topology. They provide a framework for understanding the behavior of directed graphs and the relationships between their cycles. Overall, trace monoids and hike monoids play a crucial role in advancing research in combinatorics and related areas.
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2. What are the properties of hike dependency graphs of all graphs with in-and out-degrees at least 2?
The hike dependency graphs of all graphs with in-and out-degrees at least 2 provide constraints on the number of vertices and edges that are satisfied by at least one graph in ph-1(H). These graphs are diverse and do not share basic graph theoretic properties such as regularity. However, they must share certain properties due to the isomorphism between their hike monoids. The hike dependency graph H determines the graph G uniquely up to isomorphism, except for the special case of G = {K3, K1,5}. The induced subgraph of H formed by the backtracks is the line graph L(G) of G, which determines G uniquely. There are graphs G and G with ph(G)/L(G) = ph(G)/L(G) and G and G do not share fundamental graph theoretic properties. Deciding whether there exists a graph G such that H = ph(G)/L(G) is an open problem in graph theory. These issues are relevant in network analysis and algebraic graph theory, as many methods disregard or forbid backtracks. The cycle double cover conjecture is an open problem related to these graphs and cycles.
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3. What are the combinatorial invariants sensitive to how walks and hikes are composed of cycles?
The combinatorial invariants sensitive to how walks and hikes are composed of cycles are represented by the functions f ohm H G (z) and f ohm W G (z). These functions are associated with hikes and walks, respectively, and are defined as hH G z ohm(h) and hW G z ohm(h). The invariants are preserved under isomorphism of hike monoids, as shown by the equations 1. f ohm H G (z) = f ohm H G (z), 2. f ohm W G (z) = f ohm W G (z), 3. f ohm W G:i-j (z) = f ohm W G :i -j (z), and 4. The shape of the branched continued fraction representation of the generating series associated with an additive function s on rooted walks is the same for both G and G under isomorphism. These invariants are determined by the number of simple cycles in a hike or walk and are preserved under isomorphism of hike monoids.
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4. Weak relation between digraphs and walks?
In this work, we investigated the relation between digraphs and their walks and found it to be rather weak. No graph theoretic property seems to be directly related to the arrangement of simple cycles on a digraph. Among the graph properties that may be lost while leaving the hike monoidal structure of cycles invariant are vertex-transitivity, regularity, planarity, bipartiteness, (bi)directedness, Hamiltonicity, being Eulerian, being chordal, being triangle-free, chromatic number, graph spectra, in-and-out degree distributions, and a majority of algebraic quantities computable from adjacency matrices. This list is non-exhaustive. There is a great variety of digraphs with isomorphic hike monoids, and characterizing all transformations relating such digraphs remains open. Deciding which arrangements of simple cycles exist is highly non-trivial. Both characterization and existence questions concern simple undirected graphs when information about their line graph is removed. Walks are much less dependent on the digraphs they take place in than previously thought, indicating the need for a theory of walks distinct from graph theory. We relied on hike monoids, which are plain trace monoids, but there is no simple way to know which trace monoids are hike monoids. Trace theory is not better equipped than graph theory to address the questions raised. A self-loop can be added to obtain distinct hike monoids, resulting in different digraphs.
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