PageRank is a ranking method that assigns scores to web pages using the limit distribution of a random walk on the web graph. A fibration of graphs is a morphism that is a local isomorphism of in-neighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. We show that a deep connection relates fibrations and Markov chains with restart, a particular kind of Markov chains that include the PageRank one as a special case. This fact provides constraints on the...

Let $G$ be a graph. A vertex subversion strategy of $G$, say $S$, is a set of vertices in $G$ whose closed neighborhood is removed from $G$. The survival-subgraph is denoted by $G/S$. The Neighbor-Integrity of $G$, $\mathrm{N}I\left(G\right)$, is defined to be $\mathrm{N}I\left(G\right)={min}_{S\subseteq V\left(G\right)}\left\{\right|S|+c(G/S)\}$, where $S$ is any vertex subversion strategy of $G$, and $c(G/S)$ is the maximum order of the components of $G/S$. In this paper we give some results connecting the neighbor-integrity and binary graph operations.

In this paper, we use the graphs as a tool to study nilpotent Lie algebras. It implies to set up a link betwcen graph theory and Lie theory. To do this, it is already known that every nilpotent Lie algebra of maximal rank is associated with a generalized Cartan matrix A and it ils isomorphic to a quotient of the positive part n+ of the KacMoody algebra g(A). Then, if A is affine, we can associate n+ with a directed graph (from now on, we use the term digraph) and we can also associate a subgraph...