A graph G is singular of nullity (> 0), if its adjacency matrix A is singular, with the eigenvalue zero of multiplicity . A singular graph having a 0-eigenvector, x, with no zero entries, is called a core graph.We place particular emphasis on nut graphs, namely the core graphs of nullity one. Through symmetry considerations of the automorphism group of the graph, we study relations among the en...

RNA secondary structure consists of elements such as stems, bulges, loops. The most obvious and important scalar number that can be attached to an RNA structure is its free energy, with a landscape that governs the folding pathway. However, because of the unique geometry of RNA secondary structure, another interesting single-signed scalar number based on geometrical scales exists that can assis...

It is well known that the smallest eigenvalue of the adjacency matrix of a connected ¿-regular graph is at least -d and is strictly greater than -d if the graph is not bipartite. More generally, for any connected graph G = (V JE), consider the matrix Q = D + A where D is the diagonal matrix of degrees in the graph G , and A is the adjacency matrix of G . Then Q is positive semi-definite, and th...

The Perron-Frobenius Theorem for irreducible higher order tensors does not guarantee simplicity of the unique positive eigenvalue nor does it guarantee the unique positive eigenvalue is strictly larger than the modulus of any other eigenvalue. Irreducibility of a tensor in relation to a graph is studied. A higher order tensor is essentially positive if it takes the positive cone to its interior...

We analyze the largest eigenvalue and eigenvector for the adjacency matrices of sparse random graph. Let λ1 be the largest eigenvalue of an n-vertex graph, and v1 be its corresponding normalized eigenvector. For graphs of average degree d log n, where d is a large enough constant, we show λ1 = d log n + 1 ± o(1) and 〈1, v1〉 = √ n ( 1−Θ ( 1 logn )) . It shows a limitation of the existing method ...

We consider weighted graphs, where the edge weights are positive definite matrices. The Laplacian of the graph is defined in the usual way. We obtain an upper bound on the largest eigenvalue of the Laplacian and characterize graphs for which the bound is attained. The classical bound of Anderson and Morley, for the largest eigenvalue of the Laplacian of an unweighted graph follows as a special ...

The kernel polynomial method (KPM) is a standard tool in condensed matter physics to estimate the density of states for a quantum system. We use the KPM to instead estimate the eigenvalue densities of the normalized adjacency matrices of “natural” graphs. Because natural graph spectra often include high-multiplicity eigenvalues corresponding to certain motifs in the graph, we introduce a pre-pr...

A graph, consisting of undirected edges, can be represented as a sum of two digraphs, consisting of oppositely oriented directed edges. Gutman and Plath in [J. Serb. Chem. Soc. 66 (2001), 237–241] showed that for annulenes, the eigenvalue spectrum of the graph is equal to the sum of the eigenvalue spectra of respective two digraphs. Here we exhibit a number of other graphs with this property.

Among the eigenvalue problems of the Laplacian, the biharmonic operator eigenvalue problems are interesting projects because these problems root in physics and geometric analysis. The buckling problem is one of the most important problems in physics, and many studies have been done by the researchers about the solution and the estimate of its eigenvalue. In this paper, first, we obtain the evol...