Related papers: Planarity criteria for metric graphs
We consider Schr\"odinger operators on threshold graphs and prove a formula for the Colin de Verdi\`ere parameter in terms of the building sequence. We construct an optimal Colin de Verdi\`ere matrix for each connected threshold graph $G$…
In this paper we characterize graphs which maximize the spectral radius of their adjacency matrix over all graphs of Colin de Verdi\`ere parameter at most $m$. We also characterize graphs of maximum spectral radius with no $H$ as a minor…
In this small note, we collect several observations pertaining to the famous spectral graph parameter $\mu$ introduced in 1990 by Y. Colin de Verdi\`ere. This parameter is defined as the maximum corank among certain matrices akin to…
This note discusses a relation between the multiplicity m of the second eigenvalue {\lambda}2 of a Laplacian on a graph G, tight mappings of G and a discrete analogue of Courant's nodal line theorem. For a certain class of graphs, we show…
The Colin de Verdiere number of graph G, denoted by \mu(G), is a spectral invariant of G that is related to some of its topological properties. For example, \mu(G) \leq 3 iff G is planar. A penny graph is the contact graph of equal-radii…
The second-largest eigenvalue and second-smallest Laplacian eigenvalue of a graph are measures of its connectivity. These eigenvalues can be used to analyze the robustness, resilience, and synchronizability of networks, and are related to…
We study the maximum number of edges in an $n$ vertex graph with Colin de Verdi\`{e}re parameter no more than $t$. We conjecture that for every integer $t$, if $G$ is a graph with at least $t$ vertices and Colin de Verdi\`{e}re parameter at…
The Colin de Verdi\`ere number $\mu(G)$ of a graph $G$ is the maximum corank of a Colin de Verdi\`ere matrix for $G$ (that is, of a Schr\"odinger operator on $G$ with a single negative eigenvalue). In 2001, Lov\'asz gave a construction that…
We provide a combinatorial and self-contained proof that for all graphs $G$ embedded on a surface $S$, the Colin de Verdi\`ere parameter $\mu(G)$ is upper bounded by $7-2\chi(S)$.
For a given complex square matrix $A$ with constant row sum, we establish two new eigenvalue inclusion sets. Using these bounds, first we derive bounds for the second largest and smallest eigenvalues of adjacency matrices of $k$-regular…
Motivated by discrete Laplacian differential operators with various accuracy orders in numerical analysis, we introduce new matrices attached to a simple graph that can be considered graph Laplacians with higher accuracy. In particular, we…
In this paper, we define and compare four new measures of graph irregularity. We use these measures to prove upper bounds for the chromatic number and the Colin de Verdiere parameter. We also strengthen the concise Turan theorem for…
We present a method for proving upper bounds on the eigenvalues of the graph Laplacian. A main step involves choosing an appropriate "Riemannian" metric to uniformize the geometry of the graph. In many interesting cases, the existence of…
Laplace operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of $\delta$ and $\delta'$ types. Assuming rational independence of edge lengths, necessary and sufficient…
We establish a sharp lower bound on the first non-trivial eigenvalue of the Laplacian on a metric graph equipped with natural (i.e., continuity and Kirchhoff) vertex conditions in terms of the diameter and the total length of the graph.…
For a given graph G and an associated class of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in G, the collection of all possible spectra for such matrices is considered. Building on the pioneering work…
The Fiedler value $\lambda_2$, also known as algebraic connectivity, is the second smallest Laplacian eigenvalue of a graph. We study the maximum Fiedler value among all planar graphs $G$ with $n$ vertices, denoted by $\lambda_{2\max}$, and…
Graphs with few distinct eigenvalues have been investigated extensively. In this paper, we focus on another relevant topic: characterizing graphs with some eigenvalue of large multiplicity. Specifically, the normalized Laplacian matrix of a…
We introduce and study Laplacians on a finite metric graph endowed with generalized densities, that is, measures of finite mass. One important motivation is that this setting provides a common framework for several interesting classes of…
We investigate the expected value of various graph parameters associated with the minimum rank of a graph, including minimum rank/maximum nullity and related Colin de Verdi\`ere-type parameters. Let $G(v,p)$ denote the usual…