Let G be a graph with n vertices. We denote the largest signless Laplacian eigenvalue of G by q1(G) and Laplacian eigenvalues of G by μ1(G) > · · · > μn−1(G) > μn(G) = 0. It is a conjecture on Laplacian spread of graphs that μ1(G)−μn−1(G) 6 n − 1 or equivalently μ1(G) + μ1(G) 6 2n − 1. We prove the conjecture for bipartite graphs. Also we show that for any bipartite graph G, μ1(G)μ1(G) 6 n(n − ...

In this paper we propose an inexact spectral matching algorithm that embeds large graphs on a low-dimensional isometric space spanned by a set of eigenvectors of the graph Laplacian. Given two sets of eigenvectors that correspond to the smallest non-null eigenvalues of the Laplacian matrices of two graphs, we project each graph onto its eigenenvectors. We estimate the histograms of these one-di...

We prove interlacing inequalities between spectral minimal energies of metric graphs built on Dirichlet and standard Laplacian eigenvalues, as recently introduced in [Kennedy et al, arXiv:2005.01126]. These inequalities, which involve the first Betti number degree one vertices graph, recall both other for eigenvalues whole well estimates difference nodal Neumann domains graph eigenfunctions. To...

Let G be a simple connected graph with order n. $$\mathcal{L}(G)$$ and $$\mathcal{Q}(G)$$ the normalized Laplacian signless matrices of G, respectively. λk(G) k-th smallest eigenvalue G. Denote by ρ(A) spectral radius matrix A. In this paper, we study behaviors λ2(G) $$\rho(\mathcal{L}(G))$$ when is perturbed three operations. We also properties X for bipartite graphs, where unit eigenvector co...

Write (A) = 1 (A) min (A) for the eigenvalues of a Hermitian matrix A. Our main result is: let A be a Hermitian matrix partitioned into r r blocks so that all diagonal blocks are zero. Then for every real diagonal matrix B of the same size as A; (B A) B + 1 r 1 : Let G be a nonempty graph, (G) be its chromatic number, A be its adjacency matrix, and L be its Laplacian. The above inequality impli...

In this paper, we investigate the spectral properties of the adjacency and the Laplacian matrices of random graphs. We prove that: (i) the law of large numbers for the spectral norms and the largest eigenvalues of the adjacency and the Laplacian matrices; (ii) under some further independent conditions, the normalized largest eigenvalues of the Laplacian matrices are dense in a compact interval ...

In this paper, we investigate the spectral properties of the adjacency and the Laplacian matrices of random graphs. We prove that (i) the law of large numbers for the spectral norms and the largest eigenvalues of the adjacency and the Laplacian matrices; (ii) under some further independent conditions, the normalized largest eigenvalues of the Laplacian matrices are dense in a compact interval a...

Let G = (V ,E) be a simple connected graph and λ1(G) be the largest Laplacian eigenvalue of G. In this paper, we prove that: 1. λ1(G) = max{du +mu : u ∈ V } if and only if G is a regular bipartite or a semiregular bipartite graph, where du and mu denote the degree of u and the average of the degrees of the vertices adjacent to u, respectively. 2. λ1(G) = 2 + √ (r − 2)(s − 2) if and only if G is...

Parallel to the signless Laplacian spectral theory, we introduce and develop the nonlinear spectral theory of signless 1-Laplacian on graphs. Again, the first eigenvalue μ1 of the signless 1-Laplacian precisely characterizes the bipartiteness of a graph and naturally connects to the maxcut problem. However, the dual Cheeger constant h+, which has only some upper and lower bounds in the Laplacia...

The undirected power graph of a finite group $G$, $P(G)$, is a graph with the group elements of $G$ as vertices and two vertices are adjacent if and only if one of them is a power of the other. Let $A$ be an adjacency matrix of $P(G)$. An eigenvalue $lambda$ of $A$ is a main eigenvalue if the eigenspace $epsilon(lambda)$ has an eigenvector $X$ such that $X^{t}jjneq 0$, where $jj$ is the all-one...