Combinatorics of random walks on graphs and walk-entropies: generalized Petersen and isomerization graphs

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We consider the combinatorial enumeration of random walks on graphs with emphasis on symmetric, vertex-transitive and bipartite generalized Petersen graphs containing up to 720 vertices. We enumerate self-returning and non-returning walks originating from each vertex of graphs using the matrix power algorithms. We formulate the vertex entropies, scaled unit self-return and non-return walk entropies of structures which provide measures for the combinatorial complexity of graphs. We have chosen mathematically and chemically interesting generalized Petersen graphs G(n,k) with floral symmetries, as they find several applications in dynamic stereochemistry and several other fields. These studies reveal several interesting walk patterns and walk sequences for these graphs, and paves the way for statistical studies on these chemically and mathematically interesting graphs. Moreover, walk-based vertex partitions are machine-generated from the enumerated walk n-tuple vectors, although they do not always correlate with the automorphic partitions. Hence the present study attempts to integrate statistical mechanics, graph theory, combinatorial complexity, and symmetry for large molecular and biological networks.