Per-edge triangle count controls cut size and algebraic connectivity: a local-to-global bridge via the minimum triangle cover

A central challenge in spectral graph theory is to derive global spectral properties
of a graph from local structural constraints. We address this for the
algebraic connectivity λ2(L) and a new local invariant: the minimum peredge
triangle count τ (G) = mine∈E tri(e), where tri(e) = (A2)ij counts the
triangles containing edge e = (i, j).
The main contribution is a short combinatorial lemma: if τ (G) ≥ k, then
every non-trivial cut (S, ¯ S) satisfies |∂S| ≥ k + 1. The proof identifies, for
any cut edge e = (i, j), exactly k additional distinct cut edges forced by the
common neighbours of i and j; the distinctness follows from the absence of
self-loops. The lemma is verified exhaustively (592,464 cuts, zero violations).
From this lemma, via the Cheeger isoperimetric inequality [1, 2], we derive
a lower bound on algebraic connectivity:
λ2(L) ≥ 2(τ (G) + 1)2 n2Δ3 ,
where n is the order of G and Δ its maximum degree. This is the first lower
bound on λ2 in terms of a per-edge triangle statistic. The bound is quantitatively
weak (empirical slack ≈ 500–1300×) because the Cheeger inequality
loses a factor of h(G) in the lower direction; improving the dependence on n
and Δ is formulated as an open problem.
The result is situated in the Topostability framework [3], where τ (G) ≥
1 corresponds to the absence of always-fragile (AF) edges. An immediate
corollary is that AF(G) = 0 is equivalent to 2-edge-connectivity, providing a
graph-theoretic characterisation of a Topostability edge class.
Keywords: algebraic connectivity, triangle count, local-to-global, Cheeger
inequality, cut edges, Topostability, minimum triangle cover,
2-edge-connectivity