We develop eigenvalue estimates for the Laplacians on discrete and metric graphs using different types of boundary conditions at the vertices of the metric graph. Via an explicit correspondence of the equilateral metric and discrete graph spectrum (also in the “exceptional” values of the metric graph corresponding to the Dirichlet spectrum) we carry over these estimates from the metric graph La...

The inverse eigenvalue problem of a graph G is the characterizing all lists eigenvalues real symmetric matrices whose off-diagonal pattern prescribed by adjacencies G. strong spectral property powerful tool in this problem, which identifies entries can be perturbed while controlling and preserving eigenvalues. Matrix Liberation Lemma introduced Barrett et al. 2020 advances notion to more genera...

We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schrödinger operator with an attractive -interaction of a xed strength the support of which is a star graph with nitely many edges of an equal length L 2 (0;1]. Under the constraint of xed number of the edges and xed length of them, we prove that the lowest eigenvalue is maximized by the fully sym...

We present an extremely fast graph drawing algorithm for very large graphs, which we term ACE (for Algebraic multigrid Computation of Eigenvectors). ACE exhibits an improvement of something like two orders of magnitude over the fastest algorithms we are aware of; it draws graphs of millions of nodes in less than a minute. ACE finds an optimal drawing by minimizing a quadratic energy function. T...

In a distance-regular graph, the partition with respect to distance from a vertex supports a unique eigenvector for each eigenvalue. There may be non-singleton vertex sets whose corresponding distance partition also supports eigenvectors. We consider the members of three families of distance regular graphs, the Johnson Graphs, Hamming Graphs and Complete Multipartite graphs. For each we determi...

If G is a maximal exceptional graph then either (a) G is the cone over a graph switching-equivalent to the line graph L(K8) or (b) G has K8 as a star complement for the eigenvalue −2 (or both). In case (b) it is shown how G can be constructed from K8 using intersecting families of 3-sets.

The anti-adjacency matrix of a graph is constructed from the distance by keeping each row and column only largest distances. This can be interpreted as opposite adjacency matrix, which instead in distances equal to 1. (anti-)adjacency eigenvalues are those its matrix. Employing novel technique introduced Haemers (2019) [9], we characterize all connected graphs with exactly one positive eigenval...

A method for solving the descriptor discrete-time linear system is focused. For easily, it is converted to a standard discrete-time linear system by the definition of a derivative state feedback. Then partial eigenvalue assignment is used for obtaining state feedback and solving the standard system. In partial eigenvalue assignment, just a part of the open loop spectrum of the standard linear s...

The $\lambda$-core vertices of a graph correspond to the non-zero entries some eigenvector $\lambda$ for universal adjacency matrix $\mathbf{U}$ graph. We define partition vertex set $V$ based on and its neighbourhoods at distance $r$, give number results relating structure this partition. For such partitions, we also an entropic measure information content graph, related every distinct eigenva...