Related papers: Expected Crossing Numbers
We review results on linearly edge-reinforced random walks. On finite graphs, the process has the same distribution as a mixture of reversible Markov chains. This has applications in Bayesian statistics and it has been used in studying the…
We present a simple, yet useful result about the expected value of the determinant of random sum of rank-one matrices. Computing such expectations in general may involve a sum over exponentially many terms. Nevertheless, we show that an…
In this paper, we propose a new type of graph, denoted as "embedded-graph", and its theory, which employs a distributed representation to describe the relations on the graph edges. Embedded-graphs can express linguistic and complicated…
In this paper, we first review the weighted-versiion of the handshaking lemma based on the idea of a weighted vertex-edge incidence matrix of a given graph $G$. Then, we obtain a generalized version of the handshaking lemma based on the…
We study eigenvalue distribution of the adjacency matrix $A^{(N,p, \alpha)}$ of weighted random bipartite graphs $\Gamma= \Gamma_{N,p}$. We assume that the graphs have $N$ vertices, the ratio of parts is $\frac{\alpha}{1-\alpha}$ and the…
Various kinds of spread of influence occur in real world social and virtual networks. These phenomena are formulated by activation processes and irreversible dynamic monopolies in combinatorial graphs representing the topology of the…
Research about crossings is typically about minimization. In this paper, we consider \emph{maximizing} the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured…
The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. Our main result is that every graph $G$ that does not contain a fixed graph as a minor has crossing number $O(\Delta n)$, where $G$…
Consider the complete n-vertex graph whose edge-lengths are independent exponentially distributed random variables. Simultaneously for each pair of vertices, put a constant flow between them along the shortest path. Each edge gets some…
We study the asymptotic behavior of four statistics associated with subtrees of complete graphs: the uniform probability $p_n$ that a random subtree is a spanning tree of $K_n$, the weighted probability $q_n$ (where the probability a…
In this paper, we study a regular rooted coloured tree with random labels assigned to its edges, where the distribution of the label assigned to an edge depends on the colours of its endpoints. We obtain some new results relevant to this…
It is proved that the rectilinear crossing number of every graph with bounded tree-width and bounded degree is linear in the number of vertices. **** This paper has been withdrawn by the author. **** The results have been superseeded by the…
In this article, we are going to introduce the weighted numerical range which is a further generalization both the c-numerical range and the rank k numerical range. If the boundaries of weighted numerical ranges of two matrices (possibly of…
We give estimates for the expectation of the norm of random matrices with independent but not necessarily identically distributed entries.
We conjecture that the distribution of the edge-disjoint union of two random regular graphs on the same vertex set is asymptotically equivalent to a random regular graph of the combined degree, provided it grows as the number of vertices…
We present an entropy comparison result concerning weighted sums of independent and identically distributed random variables.
We study random graphs with arbitrary distributions of expected degree and derive expressions for the spectra of their adjacency and modularity matrices. We give a complete prescription for calculating the spectra that is exact in the limit…
Let G be a vertex transitive graph. A study of the range of simple random walk on G and of its bridge is proposed. While it is expected that on a graph of polynomial growth the sizes of the range of the unrestricted random walk and of its…
We expand the theory of pebbling to graphs with weighted edges. In a weighted pebbling game, one player distributes a set amount of weight on the edges of a graph and his opponent chooses a target vertex and places a configuration of…
We prove new lower bounds on the likely size of a maximum independent set in a random graph with a given average degree. Our method is a weighted version of the second moment method, where we give each independent set a weight based on the…