A graph is said to be determined by the adjacency and Laplacian spectrum (or to be a DS graph, for short) if there is no other non-isomorphic graph with the same adjacency and Laplacian spectrum, respectively. It is known that connected graphs of index less than 2 are determined by their adjacency spectrum. In this paper, we focus on the problem of characterization of DS graphs of index less th...

In an important paper, Alon [2] derived a Cheeger–type inequality [8], by bounding from below the second smallest eigenvalue of the Laplacian of a finite undirected graph by a function of a (vertex) isoperimetric constant. More precisely, let G=(V,E) be a finite, undirected, connected graph, and let λ2(G) denote twice (for reasons explained below) the smallest non-zero eigenvalue of the Laplaci...

Gain graphs are where the edges given some orientation and labeled with elements (called gains) from a group so that gains inverted when we reverse direction of edges. Generalizing notion gain graphs, skew have property reversed edge is image under an anti-involution. In this paper, study two different types, Laplacian g -Laplacian matrices for graph taken multiplicative F x field characteristi...

Gene selection is an important research topic in pattern recognition and tumor classification. Numerous methods have been proposed, Maximum Margin Criterion (MMC) is one of the famous methods have been proposed to solve the small size samples problem. But, the MMC only considers the global structure of samples. In this article, a novel recursive gene selection criterion named Laplacian Maximum ...

Spectral techniques are often used to partition the set of vertices a graph, or form clusters. They based on Laplacian matrix. These allow easily integrate weights edges. In this work, we introduce p-Laplacian, generalized matrix with potential, which also allows us take into account vertices. vertex independent edge weights. way, can cluster importance vertices, assigning more weight some than...

One of the classical results in graph theory is the matrix-tree theorem which asserts that the determinant of a cofactor of the combinatorial Laplacian is equal to the number of spanning trees in a graph (see [1, 7, 11, 15]). The usual notion of the combinatorial Laplacian for a graph involves edge weights. Namely, a Laplacian L for G is a matrix with rows and columns indexed by the vertex set ...

In this paper, we describe a methodology for comparing networks represented as weighted graphs. The key idea is to associate a probability density function derived from the graph Laplacian, and then compute the Wasserstein distance between the derived densities of the respective graphs. This has wide applications to various networks including biological and financial.

We study partitions of the two-dimensional flat torus (R/Z) × (R/bZ) into k domains, with b a real parameter in (0, 1] and k an integer. We look for partitions which minimize the energy, defined as the largest first eigenvalue of the Dirichlet Laplacian on the domains of the partition. We are in particular interested in the way these minimal partitions change when b is varied. We present here a...