The numbers of spanning trees of the cubic cycle C3N and the quadruple cycle C4N
Discrete Mathematics
The number of spanning trees in circulant graphs
Discrete Mathematics
The Combinatorics of Network Reliability
The Combinatorics of Network Reliability
Unhooking circulant graphs: a combinatorial method for counting spanning trees and other parameters
WG'04 Proceedings of the 30th international conference on Graph-Theoretic Concepts in Computer Science
Counting spanning trees and other structures in non-constant-jump circulant graphs
ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
Unhooking circulant graphs: a combinatorial method for counting spanning trees and other parameters
WG'04 Proceedings of the 30th international conference on Graph-Theoretic Concepts in Computer Science
Critical groups of graphs with dihedral actions
European Journal of Combinatorics
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It has long been known that the number of spanning trees in circulant graphs with fixed jumps and n nodes satisfies a recurrence relation in n. The proof of this fact was algebraic (relating the products of eigenvalues of the graphs’ adjacency matrices) and not combinatorial. In this paper we derive a straightforward combinatorial proof of this fact. Instead of trying to decompose a large circulant graph into smaller ones, our technique is to instead decompose a large circulant graph into different step graph cases and then construct a recurrence relation on the step graphs. We then generalize this technique to show that the numbers of Hamiltonian Cycles, Eulerian Cycles and Eulerian Orientations in circulant graphs also satisfy recurrence relations.