arXiv subjects
Publications and source records attributed to Ian Malcolm Johnson McInnis.
The family of cycle completable graphs has several cryptomorphic descriptions, the equivalence of which has heretofore been proven by a laborious implication-cycle that detours through a motivating matrix completion problem. We give a concise proof, partially by introducing a new characterization. Then we generalize this family to ``$k$-quasichordal'' graphs, with three natural characterizations.
A weighted graph $ϕG$ encodes a finite metric space $D_{ϕG}$. When is $D$ totally decomposable? When does it embed in $\ell_1$ space? When does its representing matrix have $\leq 1$ positive eigenvalue? We give useful lemmata and prove that these questions can be answered without examining $ϕ$ if and only if $G$ has no $K_{2,3}$ minor. We also prove results toward the following conjecture. $D_{ϕG}$ has $\leq n$ positive eigenvalues for all $ϕ$, if and only if $G$ has no $K_{2,3,...,3}$ minor, with $n$ threes.
We study the inertia of distance matrices of weighted graphs. Our novel congruence-based proof of the inertia of weighted trees extends to a proof for the inertia of weighted unicyclic graphs whose cycle is a triangle. Partial results are given on the inertia of other rationally weighted unicylic graphs.