In this paper, we introduce the Hosoya-Spectral indices and the Hosoya information content of a graph. The first measure combines structural information captured by partial Hosoya polynomials and graph spectra. The latter is a graph entropy measure which is based on blocks consisting of vertices with the same partial Hosoya polynomial. We evaluate the discrimination power of these quantities by...

Electronic absorption spectra of chrysophanol (1,8-dihydroxy-3-methylanthracene-9,10-dione), physcion (1,8-dihydroxy-3-methoxy-6-methylanthracene-9,10-dione) and emodin (1,3,8trihydroxy-6-methylanthracene-9,10-dione) were investigated. Molecular geometries of the substituted anthraquinones in the ground state were optimized using semiempirical AM1 method without imposing any symmetry constrains...

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−1/2AD−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 λ = (λ1, . . . , λ|V |), 0 ≤ λ...

The Merrifield-Simmons index of a graph is defined as the total number of the independent sets of the graph and the Hosoya index of a graph is defined as the total number of the matchings of the graph. In this paper, we give formula for Merrifield-Simmons and Hosoya indices of some classes of cartesian product of two graphs K{_2}×H, where H is a path graph P{_n}, cyclic graph C{_n}, or star gra...

In this paper, we investigate the properties of the cell-graphs by using the spectral graph theory. The spectral analysis is performed on (i) the adjacency matrix of a cell-graph, and (ii) the normalized Laplacian of the cell-graph. We show that the spectra of the cell-graphs of cancerous tissues are unique and the features extracted from these spectra distinguish the cancerous (malignant gliom...

In this paper, we demonstrate how PCA and ICA can be used for embedding graphs in pattern-spaces. Graph spectral feature vectors are calculated from the leading eigenvalues and eigenvectors of the unweighted graph adjacency matrix. The vectors are then embedded in a lower dimensional pattern space using both the PCA and ICA decomposition methods. Synthetic and real sequences are tested using th...

We study here the spectra of random lifts of graphs. Let G be a finite connected graph, and let the infinite tree T be its universal cover space. If λ1 and ρ are the spectral radii of G and T respectively, then, as shown by Friedman [Fri03], in almost every n-lift H of G, all “new” eigenvalues of H are ≤ O ( λ 1/2 1 ρ 1/2 ) . Here we improve this bound to O ( λ 1/3 1 ρ 2/3 ) . It is conjectured...