bibkey: bonetto2024mixed
authors: Riccardo Bonetto; Hildeberto Jardón Kojakhmetov
year: 2024
title: "On the Eigenvalues of Graphs with Mixed Algebraic Structure"
doi: null
url: https://arxiv.org/html/2408.00487v1
claim: "Definition 3 assigns the matrix M(ε)=L_G+ε A_H to a disjoint spanning edge partition of a finite simple graph, for ε>0. Definitions 7 and 8 count real eigenvalues with multiplicity and define signature as Nplus minus Nminus. Section 3, Conjecture 1 asks whether this signature is nonincreasing in ε."
strata_touched:
  - D5/S3/Combinatorics/SpectralGraph/BonettoSignature
license: citation-only
triage: anchor

Mixed Laplacian and adjacency structure

The paper considers a finite simple undirected graph F and spanning subgraphs G,H with disjoint edge sets whose union is the edge set of F. Definitions 1 and 2 use the ordinary 0/1 adjacency matrix and the combinatorial Laplacian D−A. Definition 3 sets

$$M(u)=L_G+u{A}_H,u>0.$$

Remark 1 states that these are real symmetric matrices with a complete orthogonal eigenbasis. Definition 7 counts negative, zero and positive eigenvalues with multiplicity. Definition 8 defines

$$s(Q)=N_{+}(Q)-N_{-}(Q).$$

Conjecture 1 in Section 3 reads:

The signature of the matrix M(ε) is a monotonic decreasing function of ε.

Here decreasing is interpreted as nonincreasing: the accompanying step plots include constant intervals. The quantified assertion is

$$0<s\le t⟹s(M(t))\le s(M(s)).$$

No invertibility, connectivity, simple spectrum or small-parameter restriction is imposed on the conjecture. Singular matrices and zero eigenvalues remain part of its scope. The formal statement also includes the empty vertex type. The paragraph following the conjecture proposes the corresponding assertion for A+uB with A real symmetric positive semidefinite and B real symmetric. The subsequent limiting-signature and transition-count claims are separate assertions.

Verified locator

• URL: https://arxiv.org/html/2408.00487v1 — Definitions 1–3 and Remark 1 in Section 2; Definitions 7–8 and Remark 2; Section 3, Conjecture 1 and its following discussion and plots; Section 4, Outlook. The primary HTML was read directly. Its 310776 bytes have SHA-256 49a537fe5a5f96ce679a8f487f0e9aa4053e2f7af7e46c79dcc83c12774b4b88.

The inspected arXiv version history lists only v1, submitted 1 August 2024. That source labels the assertion Conjecture 1, and its outlook asks for a proof or further understanding. Bounded title, author and content searches supplied no later exact settlement; inaccessible and unindexed sources remain outside that finding. This is not an assertion of worldwide priority or exhaustive literature coverage. The matrix definitions and the conjecture are source material; the formal positive-subspace assembly is repository-derived from existing inertia results.