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Making Walks Count: From Silent Circles to Hamiltonian Cycles

Max A. Alekseyev, Gerard P. Michon

TL;DR

This work demonstrates the matrix-transfer method for enumerating walks, Hamiltonian cycles and paths, and fixed-length cycles in graphs, by encoding configurations into small state graphs and using adjacency-matrix powers. It develops concrete results for Silent Circles via an 8-node gaze digraph, yielding generating functions and recurrences that link circle and prism configurations; it then analyzes antiprism graphs by decomposing Hamiltonian cycles into two types and deriving exact counts through a 4-state signature graph. Extending to arbitrary graphs, the authors present inclusion-exclusion based formulas that convert Hamiltonian-path/cycle counts into sums of traces of subgraph adjacency matrices, highlighting the exponential-time nature but offering a practical, exact method. Finally, the paper generalizes to simple $k$-cycles and $k$-paths with explicit formulas and demonstrates computations on higher-dimensional graphs (e.g., 24-cell), illustrating the broad applicability of transfer-matrix techniques to graph enumeration problems.

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.

Making Walks Count: From Silent Circles to Hamiltonian Cycles

TL;DR

This work demonstrates the matrix-transfer method for enumerating walks, Hamiltonian cycles and paths, and fixed-length cycles in graphs, by encoding configurations into small state graphs and using adjacency-matrix powers. It develops concrete results for Silent Circles via an 8-node gaze digraph, yielding generating functions and recurrences that link circle and prism configurations; it then analyzes antiprism graphs by decomposing Hamiltonian cycles into two types and deriving exact counts through a 4-state signature graph. Extending to arbitrary graphs, the authors present inclusion-exclusion based formulas that convert Hamiltonian-path/cycle counts into sums of traces of subgraph adjacency matrices, highlighting the exponential-time nature but offering a practical, exact method. Finally, the paper generalizes to simple -cycles and -paths with explicit formulas and demonstrates computations on higher-dimensional graphs (e.g., 24-cell), illustrating the broad applicability of transfer-matrix techniques to graph enumeration problems.

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.

Paper Structure

This paper contains 4 sections, 6 theorems, 19 equations, 4 figures.

Table of Contents

  1. Silent Circles
  2. Hamiltonian Cycles in Antiprism Graphs
  3. Hamiltonian Cycles and Paths in Arbitrary Graphs
  4. Simple Cycles and Paths of a Fixed Length

Key Result

Theorem 1

The number $(A^n)_{i,j}$ equals the number of walks of length $n$ going from node $i$ to node $j$ in the digraph with the adjacency matrix $A$.

Figures (4)

  • Figure 1: The gaze digraph and its adjacency matrix $A$.
  • Figure 2: (a) The antiprism graph $C_{10}^{1,2}$. (b) A directed Hamiltonian cycle in $C_{10}^{1,2}$ that does not visit either of the edges $(4,6)$, $(5,6)$, $(5,7)$, i.e., has signature $000$ at node 4.
  • Figure 3: Possible visitation signatures for a Hamiltonian cycle of type (T2) in $C_{2n}^{1,2}$. Visited and non-visited edges are shown as solid and dashed, respectively.
  • Figure 4: The signature graph $S$ and its adjacency matrix $B$.

Theorems & Definitions (9)

  • Theorem 1: stanley
  • Corollary 2
  • Theorem 3: stanley
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • proof