We survey results concerning Hamilton cycles in random graphs. Specifically, we focus on existence results for general and regular graphs, and discuss algorithms for finding Hamilton cycles and solving related problems (that succeed with high probability).

We introduce the concept of pattern graphs–directed acyclic graphs representing how response patterns are associated. A graph represents an identifying restriction that is nonparametrically identified/saturated and often a missing not at random restriction. selection model mixture formulations using show they equivalent. leads to inverse probability weighting estimator as well imputation-based ...

We consider three models of sparse random graphs: undirected and directed Erdős–Rényi graphs bipartite graph with two equal parts. For such graphs, we show that if the edge connectivity probability p satisfies $$np\ge \log n+k(n)$$ $$k(n)\rightarrow \infty $$ as $$n\rightarrow , then adjacency matrix is invertible approaching one (n number vertices in former cases same for each part latter case...

We give a new proof of the classical Erdös theorem on the existence of graphs with arbitrarily high chromatic number and girth. Rather than considering random graphs where the edges are chosen with some carefully adjusted probability, we use a simple counting argument on a set of graphs with bounded vertex degree.

Suppose we are given two graphs on n vertices. We define an observable in the Hilbert space C[(Sn ≀ S2)m] which returns the answer “yes” with certainty if the graphs are isomorphic and “no” with probability at least 1 − n! 2 if the graphs are not isomorphic. We do not know if this observable is efficiently implementable.

We consider the Dirac equation on periodic networks (quantum graphs). The self-adjoint quasi boundary conditions are derived. secular allowing us to find energy spectrum of particles quantum graphs is obtained. Band spectra different topologies calculated. Universality probability be in for certain graph observed.

1.1 Random (ER) graphs The Erdös-Rényi (ER) random graphs model, also called simply random graphs, was presented by Erdös and Rényi [4] in the 1950s and 1960s. Erdös and Rényi characterized random graphs and showed that many of the properties of such networks can be calculated analytically. Construction of an ER random graph with parameter 0 ≤ p ≤ 1 and N nodes is by connecting every pair of no...

‎in this paper we introduce mixed unitary cayley graph $m_{n}$‎ ‎$(n>1)$ and compute its eigenvalues‎. ‎we also compute the energy of‎ ‎$m_{n}$ for some $n$‎.

We consider a sequence of random graphs constructed by a hierarchical procedure. The construction replaces existing edges by pairs of edges in series or parallel with probability p and 1−p respectively. We investigate the effective resistance across the graphs, first-passage percolation on the graphs and the Cheeger constants of the graphs as the number of edges tends to infinity. In each case ...