Tamara Koledin

Sharp spectral inequalities for connected bipartite graphs with maximal Q-index

The Q -index of a simple graph is the largest eigenvalue of its signless Laplacian. As for the adjacency spectrum, we will show that in the set of connected bipartite graphs with fixed order and size, the bipartite graphs with maximal Q -index are the double nested graphs. We provide a sequence of (in)equalities regarding the principal eigenvector of the signless Laplacian of double nested graphs and apply these results to obtain some lower and upper bounds for their Q -index. In the end, we give some computational results in order to compare these bounds.

Connected signed graphs of fixed order, size, and number of negative edges with maximal index

Abstract In this paper we focus on connected signed graphs of fixed number of vertices, positive edges and negative edges that maximize the largest eigenvalue (also called the index) of their adjacency matrix. In the first step we determine these signed graphs in the set of signed generalized theta graphs. Concerning the general case, we use the eigenvector techniques for getting some structural properties of resulting signed graphs. In particular, we prove that positive edges induce nested split subgraphs, while negative edges induce double nested signed subgraphs. We observe that our concept can be applied when considering balancedness of signed graphs (the property that is extensively studied in both mathematical and non-mathematical context).

Trees with small spectral gap

Continuing the previous research, we consider trees with given number of vertices and minimal spectral gap. Using the computer search, we conjecture that this spectral invariant is minimized for double comet trees. The conjecture is confirmed for trees with at most 20 vertices; simultaneously no counterexamples are encountered. We provide theoretical results concerning double comets and putative trees that minimize the spectral gap. We also compare the spectral gap of regular graphs and paths. Finally, a sequence of inequalities that involve the same invariant is obtained.

Methods for Limiting the Calculation Area During Problem Solving by the Finite Difference Method

By using the integro-differential approach and classical boundary conditions (such as Dirichlets, Neumanns or the very rarely used Cauchy boundary condition) for solving the two-dimensional problems in open space by the finite difference method, it is possible to - in the numerically exact way - close the calculation area to finite distance. Thus, one of great limitations of the finite difference method is overcome.