A lower bound for nodal count on discrete and metric graphs

According to a well-know theorem by Sturm, a vibrating string is divided into exactly N nodal intervals by zeros of its N-th eigenfunction. Courant showed that one half of Sturm's theorem for the strings applies to the theory of membranes: N-th eigenfunction cannot have more than N domains. He also gave an example of a eigenfunction high in the spectrum with a minimal number of nodal domains, thus excluding the existence of a non-trivial lower bound. An analogue of Sturm's result for discretizations of the interval was discussed by Gantmacher and Krein. The discretization of an interval is a graph of a simple form, a chain-graph. But what can be said about more complicated graphs? It has been known since the early 90s that the nodal count for a generic eigenfunction of the Schrodinger operator on quantum trees (where each edge is identified with an interval of the real line and some matching conditions are enforced on the vertices) is exact too: zeros of the N-th eigenfunction divide the tree into exactly N subtrees. We discuss two extensions of this result in two directions. One deals with the same continuous Schrodinger operator but on general graphs (i.e. non-trees) and another deals with discrete Schrodinger operator on combinatorial graphs (both trees and non-trees). The result that we derive applies to both types of graphs: the number of nodal domains of the N-th eigenfunction is bounded below by N-L, where L is the number of links that distinguish the graph from a tree (defined as the dimension of the cycle space or the rank of the fundamental group of the graph). We also show that if it the genericity condition is dropped, the nodal count can fall arbitrarily far below the number of the corresponding eigenfunction.Comment: 15 pages, 4 figures; Minor corrections: added 2 important reference

Bridging Elementary Landscapes and a Geometric Theory of Evolutionary Algorithms: First Steps

This is the author accepted manuscript. The final version is available from Springer via the DOI in this record.Paper to be presented at the Fifteenth International Conference on Parallel Problem Solving from Nature (PPSN XV), Coimbra, Portugal on 8-12 September.Based on a geometric theory of evolutionary algorithms, it was shown that all evolutionary algorithms equipped with a geometric crossover and no mutation operator do the same kind of convex search across representations, and that they are well matched with generalised forms of concave fitness landscapes for which they provably find the optimum in polynomial time. Analysing the landscape structure is essential to understand the relationship between problems and evolutionary algorithms. This paper continues such investigations by considering the following challenge: develop an analytical method to recognise that the fitness landscape for a given problem provably belongs to a class of concave fitness landscapes. Elementary landscapes theory provides analytic algebraic means to study the landscapes structure. This work begins linking both theories to better understand how such method could be devised using elementary landscapes. Examples on well known One Max, Leading Ones, Not-All-Equal Satisfiability and Weight Partitioning problems illustrate the fundamental concepts supporting this approach

The first Dirichlet eigenvalue of bicyclic graphs

summary:In this paper, we have investigated some properties of the first Dirichlet eigenvalue of a bicyclic graph with boundary condition. These results can be used to characterize the extremal bicyclic graphs with the smallest first Dirichlet eigenvalue among all the bicyclic graphs with a given graphic bicyclic degree sequence with minor conditions. Moreover, the extremal bicyclic graphs with the smallest first Dirichlet eigenvalue among all the bicycle graphs with fixed $k$ interior vertices of degree at least 3 are obtained