The spectrum of the normalized Laplacian matrix cannot determine the number of edges in a graph, however finding constructions of cospectral graphs with differing number of edges has been elusive. In this paper we use basic properties of twins and scaling to show how to construct such graphs. We also give examples of families of graphs which are cospectral with a subgraph for the normalized Lap...

Several researchers have recently explored various graph parameters that can or cannot be characterized by the spectrum of a matrix associated with graph. In this paper, we show several NP-hard zero forcing numbers are not spectra types matrices particular, consider standard forcing, positive semidefinite and skew provide constructions infinite families pairs cospectral graphs, which different ...

A spanning tree of graph G is a spanning subgraph of G that is a tree. In this paper, we focus our attention on (n,m) graphs, where m = n, n + 1, n + 2, n+3 and n + 4. We also determine some coefficients of the Laplacian characteristic polynomial of fullerene graphs.

Let G be a graph on n vertices. Its Laplacian matrix is the n-by-n matrix L(G) = D(G) A(G), where A(G) is the familiar (0, 1) adjacency matrix, and D(G) is the diagonal matrix of vertex degrees. This is primarily an expository article surveying some of the many results known for Laplacian matrices. Its six sections are: Introduction, The Spectrum, The Algebraic Connectivity, Congruence and Equi...

There are a lot of researches on the spectrum of the discrete Laplacian on an infinite graph in various areas. One of main topics among them is to characterize the spectral structure in terms of a certain geometric property of the graph. We focus on the possibility of the absence of eigenvalues and give a criterion for this in terms of combinatorial property of a graph. A graph G is a connected...

With ever growing databases containing multimedia data, indexing has become a necessity to avoid a linear search. We propose a novel technique for indexing multimedia databases in which entries can be represented as graph structures. In our method, the topological structure of a graph as well as that of its subgraphs are represented as vectors whose components correspond to the sorted laplacian...

Let G = (V,E) be a graph without loops and multiple edges. Let n and m be the number of vertices and edges of G, respectively. Such a graph will be referred to as an (n,m)-graph. For v ∈ V (G), let d(v) be the degree of v. In this paper, we are concerned only with undirected simple graphs (loops and multiple edges are not allowed). Let G be a graph with n vertices and the adjacency matrix A(G)....

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In...

Content based music retrieval opens up large collections, both for the general public and music scholars. It basically enables the user to find (groups of) similar melodies, thus facilitating musicological research of many kinds. We present a graph spectral approach, new to the music retrieval field, in which melodies are represented as graphs, based on the intervals between the notes they are ...