The Moran process models the spread of genetic mutations through a population. A mutant with relative fitness r is introduced into a population and the system evolves, either reaching fixation (in which every individual is a mutant) or extinction (in which none is). In a widely cited paper (Nature, 2005), Lieberman, Hauert and Nowak generalize the model to populations on the vertices of graphs....

Random graphs with high probability satisfy the n-e.c. adjacency property, although until recently few explicit examples of such graphs were known. We supply a new general construction for generating infinite familes of finite regular n-e.c. graphs derived from certain resolvable Steiner 2-designs. We supply an extension of our construction to the infinite case, and thereby give a new represent...

We study infinite limits of graphs generated by the duplication model for biological networks. We prove that with probability 1, the sole nontrivial connected component of the limits is unique up to isomorphism. We describe certain infinite deterministic graphs which arise naturally from the model. We characterize the isomorphism type and induced subgraph structure of these infinite graphs usin...

In this paper, we investigate continuous-time quantum walk on star graphs. It is shown that quantum central limit theorem for a continuous-time quantum walk on star graphs for N -fold star power graph, which are invariant under the quantum component of adjacency matrix, converges to continuous-time quantum walk on K2 graphs (Complete graph with two vertices) and the probability of observing wal...

We prove new theoretical results about several variations of the cop and robber game on graphs. First, we consider a variation of the cop and robber game which is more symmetric called the cop and killer game. We prove for all c < 1 that almost all random graphs are stalemate for the cop and killer game, where each edge occurs with probability p such that 1 nc ≤ p ≤ 1 − 1 nc . We prove that a g...

We present recent advances in harmonic analysis on infinite graphs. Our approach combines combinatorial tools with new results from the theory of unbounded Hermitian operators Hilbert space, geometry, boundary constructions, and spectral invariants. focus particular classes graphs, including such weighted graphs which arise electrical network models, as well diagrammatic graph representations. ...

We introduce I/O-effiient certifying algorithms for bipartite graphs, as well the classes of split, threshold, chain, and trivially perfect graphs. When input graph is a member respective class, algorithm returns certificate that characterizes this class. Otherwise, it forbidden induced subgraph non-membership. On with n vertices m edges, our take $$O( {\textsc {sort}(n + m)} ) $$ I/Os in worst...