Abstract
Let $c_n$ denote the number of vertex-labeled connected graphs on $n$ vertices. Using group actions and elementary number theory, we show that the infinite sequence, $c_n : n \ge 1,$ is ultimately periodic modulo every positive integer. We state and prove our results for sequences defined by a weighted generalization of $c_n$ and conjecture that these results are suggestive of similar periodic behavior of the Tutte polynomial evaluations of the complete graph $K_n$ at integer points.
Highlights
Let Cn be the set of labeled connected graphs on the n-vertex set {1, 2, . . . , n}
While we know of no simple closed form for Cn(2), there are some useful recurrence relations
Let D be the set of labeled connected graphs on the (n + pk−1)-vertex set P ∪ N
Summary
Let Cn be the set of labeled connected graphs on the n-vertex set {1, 2, . . . , n}. For b ∈ Z, we define the weighted sum (1) Cn(b) d=ef.(b − 1)|G|−n+1, G∈Cn where |G| denotes the number of edges of graph G. Our proof of Proposition 1 uses permutation group actions on sets of labeled connected graphs on n + pk vertices to obtain the recurrence congruences, where p is a prime, k ≥ 1, and n ≥ 0. Let D be the set of labeled connected graphs on the (n + pk−1)-vertex set P ∪ N .
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