We describe the asymptotic properties of the edge-triangle exponential random graph model as the natural parameters diverge along straight lines. We show that as we continuously vary the slopes of these lines, a typical graph drawn from this model exhibits quantized behavior, jumping from one complete multipartite graph to another, and the jumps happen precisely at the normal lines of a polyhed...

In this paper we propose a multipartite version of the classical Turán problem of determining the minimum number of edges needed for an arbitrary graph to contain a given subgraph. As it turns out, here the non-trivial problem is the determination of the minimal edge density between two classes that implies the existence of a given subgraph. We determine the critical edge density for trees and ...

Suppose a graph G have n vertices, m edges, and t triangles. Letting λn (G) be the largest eigenvalue of the Laplacian of G and μn (G) be the smallest eigenvalue of its adjacency matrix, we prove that λn (G) ≥ 2m2 − 3nt m (n2 − 2m) , μn (G) ≤ 3n3t− 4m3 nm (n2 − 2m) , with equality if and only if G is a regular complete multipartite graph. Moreover, if G is Kr+1-free, then λn (G) ≥ 2mn (r − 1) (...

A (di)graph is supereulerian if it contains a spanning, connected, eulerian sub(di)graph. This property is a relaxation of hamiltonicity. Inspired by this analogy with hamiltonian cycles and by similar results in supereulerian graph theory, we give a number of sufficient Ore type conditions for a digraph to be supereulerian. Furthermore, we study the following conjecture due to Thomassé and the...

The Smithsonian/NASA Astrophysics Data System exists at the nexus of a dense system of interacting and interlinked information networks. The syntactic and the semantic content of this multipartite graph structure can be combined to provide very specific research recommendations to the scientist/user.

It is shown that if K is any regular complete multipartite graph of even degree, and F is any bipartite 2-factor of K, then there exists a factorisation of K into F ; except that there is no factorisation of K6,6 into F when F is the union of two disjoint 6-cycles.

In a network of spin 12 particles, controlled through an external electro-magnetic field, the gyromagnetic ratio each is parameter that characterizes interaction with control field. Multipartite networks are such spins divided into subsets according to their and in one set interact same way all another set. Due presence symmetries these types systems, underlying Hilbert state space splits invar...

We study perfect state transfer in Grover walks, which are typical discrete-time quantum walk models. In particular, we focus on states associated to vertices of a graph. call such vertex type states. Perfect between can be studied via Chebyshev polynomials. derive necessary condition eigenvalues graph for occur. addition, perfectly determine the complete multipartite graphs whose partite sets ...

Jin and Liu discovered an elegant formula for the number of rooted spanning forests in the complete bipartite graph a1;a2 , with b1 roots in the rst vertex class and b2 roots in the second vertex class. We give a simple proof to their formula, and a generalization for complete m-partite graphs, using the multivariate Lagrange inverse. Y. Jin and C. Liu [3] give a formula for f(m; l;n; k), the n...