Tight lower bound on the intersection number of connected graphs

Prove that for every connected graph G with n = |V| and cyclomatic number ν, letting integers q and r satisfy the integer division 2ν = q(n−1) + r, the intersection number satisfies (n−1)·binom(q, 2) + q·r ≤ cap(G).

Background

The paper introduces the intersection number cap(G) associated with a spanning tree’s cycle basis, motivated by sparsity considerations of cycle intersection matrices that appear in linear systems for integrating discrete 1-forms. A general lower bound l_{n,m} = 1/2(ν^2/(n−1) − ν) is proved, and experimental evidence suggests a stronger, tight lower bound.

The authors propose an improved bound parameterized by q and r from the integer division of 2ν by n−1, reflecting an approximate equidistribution of cycle-edges across bonds tied to spanning tree edges. Establishing this bound would sharpen guarantees for sparsity and provide a universally tight benchmark across arbitrary connected graphs.