The Neumann points of an eigenfunction f on a quantum (metric) graph are the interior zeros $$f'$$ . domains sub-graphs bounded by points. and counterparts well-studied nodal domains. We prove bounds number properties probability distribution this number. Two basic presented: wavelength capacity spectral position. state those as well key features their distributions. To rigorously investigate p...

bandwidth labelling is a well known research area in graph theory. we provide a new proof that the bandwidth of mobius ladder is 4, if it is not a $k_{4}$, and investigate the bandwidth of a wider class of mobius graphs of even strips.

We look at the minimal size of a maximal matching in general, bipartite and d-regular random graphs. We prove that the ratio between the sizes of any two maximal matchings approaches one in dense random graphs and random bipartite graphs. Weaker bounds hold for sparse random graphs and random dregular graphs. We also describe an algorithm that with high probability finds a matching of size stri...

The main issue of this work is to investigate asymmetric structures in graphs. While symmetry structures in graphs are well observed, the opposite question has not been investigated deeply so far. It is known from a theorem of Wright, that almost all graphs are asymmetric. The class of asymmetric graphs is restricted further by forbidding even local symmetries. The main question is to determine...

one of the generalizations of the wiener number to weighted graphs is to assign probabilities to edges, meaning that in nonstatic conditions the edge is present only with some probability. the reliability wiener number is defined as the sum of reliabilities among pairs of vertices, where the reliability of a pair is the reliability of the most reliable path. closed expressions are derived for t...

Ecological factors exert a range of effects on the dynamics of the evolutionary process. A particularly marked effect comes from population structure, which can affect the probability that new mutations reach fixation. Our interest is in population structures, such as those depicted by 'star graphs', that amplify the effects of selection by further increasing the fixation probability of advanta...

Consider a uniform expanders family Gn with a uniform bound on the degrees. It is shown that for any p and c > 0, a random subgraph of Gn obtained by retaining each edge, randomly and independently, with probability p, will have at most one cluster of size at least c|Gn|, with probability going to one, uniformly in p and the size of the expander. The method from Ajtai, Komlós and Szemerédi [1] ...

We study the stochastic birth-death process in a finite and structured population and analyze how the fixation probability of a mutant depends on its initial placement. In particular, we study how the fixation probability depends on the degree of the vertex where the mutant is introduced, and which vertices are its neighbours. We find that within a fixed graph, the fixation probability of a mut...