In this paper, as a generalization of the binomial random graph model, we define the model of multigraphs as follows: let G( n; {p(k)}) be the probability space of all the labelled loopless multigraphs with vertex set V = {v(1), v(2), ..., v(n)}, in which the distribution of t(vi), v(j), the number of the edges between any two vertices v(i) and v(j) is P{t(vi), v(j) = k} = p(k), k = 0, 1, 2, ... and they are independent of each other. Denote by X-d = X-d(G), Y-d = Y-d(G), Z(d) = Z(d)(G) and Z(cd) = Z(cd)(G) the number of vertices of G with degree d, at least d, at most d and between c and d. In this paper, we discuss the distribution of X-d, Y-d, Z(d) and Z(cd) in the probability space G(n; {p(k)}).National Natural Science Fund of China [10831001, 10871046, 10971027]; Science and Technology of Science Fund of Fujian Province [A0950059]; Fuzhou University [2009-XQ-27
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.