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A generating function for reciprocal binomial coefficients is written down, integral representations of this function are obtained, generating functions for sums of reciprocal binomial coefficients are derived, new identities are obtained,…
In a randomly oriented graph containing vertices $x$ and $y$, denote by $\{x\to y\}$ the event that there is a directed path from $x$ to $y$. We study the correlation between the events $\{x\to y\}$ and $\{y\to z\}$ for a (large) oriented…
We prove that in a random tournament the events $\{s\rightarrow a\}$ and $\{t\rightarrow b\}$ are positively correlated, for distinct vertices $a,s,b,t \in K_n.$ It is also proven that the correlation between the events $\{s\rightarrow a\}$…
Spatially embedded networks are important in several disciplines. The prototypical spatial net- work we assume is the Random Geometric Graph of which many properties are known. Here we present new results for the two-point degree…
We investigate the threshold probability for connectivity of sparse graphs under weak assumptions. As a corollary this completely solve the problem for Cartesian powers of arbitrary graphs. In detail, let $G$ be a connected graph on $k$…
In this article we introduce and study the intersection graph of graded ideals of graded rings. The intersection graph of $G-$graded ideals of a graded ring $(R,G)$ is a simple graph, denoted by $Gr_G(R)$, whose vertices are the nontrivial…
In this paper we study cycles in random bipartite graph $G(n,n,p)$. We prove that if $p\gg n^{-2/3}$, then $G(n,n,p)$ a.a.s. satisfies the following. Every subgraph $G'\subset G(n,n,p)$ with more than $(1+o(1))n^2p/2$ edges contains a cycle…
In this paper, we give an analytic solution for graphs with n nodes and E edges for which the probability of obtaining a given graph G is specified in terms of the degree sequence of G. We describe how this model naturally appears in the…
The neighborhood polynomial of graph $G$ is the generating function for the number of vertex subsets of $G$ of which the vertices have a common neighbor in $G$. In this paper, we investigate the behavior of this polynomial under several…
Degree distribution, or equivalently called degree sequence, has been commonly used to be one of most significant measures for studying a large number of complex networks with which some well-known results have been obtained. By contrast,…
We investigate the genus $g(n,m)$ of the Erd\H{o}s-R\'enyi random graph $G(n,m)$, providing a thorough description of how this relates to the function $m=m(n)$, and finding that there is different behaviour depending on which `region' $m$…
We consider the quantity $P(G)$ associated with a graph $G$ that is defined as the probability that a randomly chosen subtree of $G$ is spanning. Motivated by conjectures due to Chin, Gordon, MacPhee and Vincent on the behaviour of this…
The conceptions of $G$-parking functions and $G$-multiparking functions were introduced in [15] and [12] respectively. In this paper, let $G$ be a connected graph with vertex set $\{1,2,...,n\}$ and $m\in V(G)$. We give the definition of…
We demonstrate how to generalize two of the most well-known random graph models, the classic random graph, and random graphs with a given degree distribution, by the introduction of hidden variables in the form of extra degrees of freedom,…
We define a growing model of random graphs. Given a sequence of nonnegative integers $\{d_n\}_{n=0}^\infty$ with the property that $d_i\leq i$, we construct a random graph on countably infinitely many vertices $v_0,v_1\ldots$ by the…
In this paper, we introduce a new concept namely degree polynomial for vertices of a simple graph. This notion leads to a concept namely degree polynomial sequence which is stronger than the concept of degree sequence. After obtaining the…
We study a random graph $G_n$, which combines aspects of geometric random graphs and preferential attachment. The resulting random graphs have power-law degree sequences with finite mean and possibly infinite variance. In particular, the…
Computation of the probability that a random graph is connected is a challenging problem, so it is natural to turn to approximations such as Monte Carlo methods. We describe sequential importance resampling and splitting algorithms for the…
One of the most influential recent results in network analysis is that many natural networks exhibit a power-law or log-normal degree distribution. This has inspired numerous generative models that match this property. However, more recent…
The paper consider an equivalence relation in the set of vertices of a bipartite graph. Some numerical characteristics showing the cardinality of equivalence classes are introduced. A combinatorial identity that is in relationship to these…