Making Walks Count: From Silent Circles to Hamiltonian Cycles

TL;DR

This paper demonstrates the matrix-transfer method's effectiveness in solving various enumeration problems related to Silent Circles, Hamiltonian cycles, and fixed-length paths in graphs.

Contribution

It introduces the application of the matrix-transfer method to new enumeration problems in graph theory and combinatorics.

Findings

Successfully enumerates Silent Circles configurations.

Counts Hamiltonian cycles in antiprism graphs.

Determines the number of fixed-length paths and cycles in arbitrary graphs.

Abstract

We illustrate the application of the matrix-transfer method for a number of enumeration problems concerning the party game Silent Circles, Hamiltonian cycles in the antiprism graphs, and simple paths and cycles of a fixed length in arbitrary graphs.