Eigenvalue bounds of the Kirchhoff Laplacian

TL;DR

The paper derives sharp upper bounds $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\$, we present 7 structured summaries for each section, a concise overarching summary, and high-signal keywords. The results rely on the paper’s core contributions: an upper bound $\\\\lambda_k \\\\le d_k+d_{k-1}$ for all $k$, a nonnegative spectrum, and a Brouwer–Haemers-type lower bound under no-multiple-connections, all grounded in the factorization $K=F^T F$ and interlacing arguments.

Abstract

We prove that each eigenvalue l(k) of the Kirchhoff Laplacian K of a graph or quiver is bounded above by d(k)+d(k-1) for all k in {1,...,n}. Here l(1),...,l(n) is a non-decreasing list of the eigenvalues of K and d(1),..,d(n) is a non-decreasing list of vertex degrees with the additional assumption d(0)=0. We also prove that in general the weak Brouwer-Haemers lower bound d(k) + (n-k) holds for all eigenvalues l(k) of the Kirchhoff matrix of a quiver.

Figures (7)

• Figure 1: The Königsberg bridge graph$G$ considered by Leonhard Euler Euler1736 in 1736 is a multi-graph of Euler characteristic $|V|-|E|=4-6=-2$ (when seen as a 1-dimensional cell complex). Removing the first row and column of the Kirchhoff matrix is the Kirchhoff matrix of a quiver with $3$ vertices.
• Figure 2: For the clover, with $K=[m]$, Theorems 1,2 are sharp. For graphs with multiple connections like the ribbon, the lower bound Theorem 2 can fail. If in the Kronecker quiver $Q=\{ V=\{V,E\}, E=\{ (V,V),(E,E),(V,E),(V,E)\} \}$, edges are interpreted as morphisms, (loops are endomorphisms), and a composition is added, one has a finite category with 2 objects. A quiver can be seen as a functor from $Q$ to the category of finite sets. Seen as such, quivers are a finite topos.
• Figure 3: The picture shows the Good-Will-Hunting quiver and the principal sub-quiver obtained by deleting the vertex $v$ with maximal degree, snapping all connections $(v,w)$ to $v$ to loops $(w,w)$ on its neighbors $w$.
• Figure 4: We see a random quiver in which the spectrum is compared with the upper bound and lower bound according to Theorem 1 and Theorem 2. The lower bound does not hold for all quivers, as Theorem 2 requires to have no multiple connections.
• Figure 5: This figure shows examples of spectra of simple graphs and compares them with known upper and lower bounds. We see first the Star, Cycle and Path graph, then the Wheel, Complete, Bipartite graph, and finally the Petersen, Grid and Random graph, all with $10$ vertices. The last is a graph with loops. The eigenvalues are outlined thick. The graph above the spectrum is upper bound of Theorem 1. The lower bound of Theorem 2 and the Horn-Johnson bounds are below. They can complement each other.

Theorems & Definitions (15)

• Theorem 1
• Lemma 1
• Theorem 2
• Lemma 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof