The Laplacian eigenvalues of graphs: a survey

Let $m_G(I)$ denote the number of Laplacian eigenvalues of a graph $G$ in an interval $I$, and let $\gamma(G)$ denote its domination number. We extend the recent result $m_G[0,1) \leq \gamma(G)$, and show that isolate-free graphs also satisfy $\gamma(G) \leq m_G[2,n]$. In pursuit of better understanding Laplacian eigenvalue distribution, we find applications for these inequalities. We relate these spectral parameters with the approximability of $\gamma(G)$, showing that $\fra...

We give inequalities relating the eigenvalues of the adjacency matrix and the Laplacian of a graph, and its minimum and maximum degrees. The results are applied to derive new conditions for quasi-randomness of graphs.

We review the properties of eigenvectors for the graph Laplacian matrix, aiming at predicting a specific eigenvalue/vector from the geometry of the graph. After considering classical graphs for which the spectrum is known, we focus on eigenvectors that have zero components and extend the pioneering results of Merris (1998) on graph transformations that preserve a given eigenvalue $\lambda$ or shift it in a simple way. These transformations enable us to obtain eigenvalues/vect...

Here we have investigated a few properties of the eigenvalues of normalized (geometric) graph Laplacian in different graphs. Preservation of eigenvalue 1 from a particular subgraph to the entire graph, the spectrum of the graph constructed with triangles share a common vertex have been addressed. Further using the number and degrees of common neighbors between vertices some new upper bounds for the largest eigenvalue have been introduced.

In this note we give a new upper bound for the Laplacian eigenvalues of an unweighted graph. Let $G$ be a simple graph on $n$ vertices. Let $d_{m}(G)$ and $\lambda_{m+1}(G)$ be the $m$-th smallest degree of $G$ and the $m+1$-th smallest Laplacian eigenvalue of $G$ respectively. Then $ \lambda_{m+1}(G)\leq d_{m}(G)+m-1 $ for $\bar{G} \neq K_{m}+(n-m)K_1 $. We also introduce upper and lower bound for the Laplacian eigenvalues of weighted graphs, and compare it with the special ...

The spectrum of a network or graph $G=(V,E)$ with adjacency matrix $A$, consists of the eigenvalues of the normalized Laplacian $L= I - D^{-1/2} A D^{-1/2}$. This set of eigenvalues encapsulates many aspects of the structure of the graph, including the extent to which the graph posses community structures at multiple scales. We study the problem of approximating the spectrum $\lambda = (\lambda_1,\dots,\lambda_{|V|})$, $0 \le \lambda_1,\le \dots, \le \lambda_{|V|}\le 2$ of $G...

Spectral graph theory is a captivating area of graph theory that employs the eigenvalues and eigenvectors of matrices associated with graphs to study them. In this paper, we present a collection of $20$ topics in spectral graph theory, covering a range of open problems and conjectures. Our focus is primarily on the adjacency matrix of graphs, and for each topic, we provide a brief historical overview.

We use two variational techniques to prove upper bounds for sums of the lowest several eigenvalues of matrices associated with finite, simple, combinatorial graphs. These include estimates for the adjacency matrix of a graph and for both the standard combinatorial Laplacian and the renormalized Laplacian. We also provide upper bounds for sums of squares of eigenvalues of these three matrices. Among our results, we generalize an inequality of Fiedler for the extreme eigenval...

Let $G$ be a graph, and let $\lambda(G)$ denote the smallest eigenvalue of $G$. First, we provide an upper bound for $\lambda(G)$ based on induced bipartite subgraphs of $G$. Consequently, we extract two other upper bounds, one relying on the average degrees of induced bipartite subgraphs and a more explicit one in terms of the chromatic number and the independence number of $G$. In particular, motivated by our bounds, we introduce two graph invariants that are of interest on...

We improve recent results relating graph eigenvalues to other graph parameters like girth, domination number, and minimum degree.