On Eccentric Digraphs of graphs

The eccentricity e(u) of a vertex u is the maximum distance of u to any other vertex of G. A vertex v is an eccentric vertex of vertex u if the distance from u to v is equal to e(u). The eccentric digraph ED(G) of a graph (digraph) G is the digraph that has the same vertex as G and an arc from u to v exists in ED(G) if and only if v is an eccentric vertex of u in G. In this paper, we have considered an open problem. Partly we have characterized graphs with specified maximum degree such that ED(G) =

Characterization of eccentric digraphs

AbstractThe eccentric digraph ED(G) of a digraph G represents the binary relation, defined on the vertex set of G, of being ‘eccentric’; that is, there is an arc from u to v in ED(G) if and only if v is at maximum distance from u in G. A digraph G is said to be eccentric if there exists a digraph H such that G=ED(H). This paper is devoted to the study of the following two questions: what digraphs are eccentric and when the relation of being eccentric is symmetric.We present a characterization of eccentric digraphs, which in the undirected case says that a graph G is eccentric iff its complement graph G¯ is either self-centered of radius two or it is the union of complete graphs. As a consequence, we obtain that all trees except those with diameter 3 are eccentric digraphs. We also determine when ED(G) is symmetric in the cases when G is a graph or a digraph that is not strongly connected

A moonshine dialogue in mathematical physics

Phys and Math are two colleagues at the University of Sa{\c c}enbon (Crefan Kingdom), dialoguing about the remarkable efficiency of mathematics for physics. They talk about the notches on the Ishango bone, the various uses of psi in maths and physics, they arrive at dessins d'enfants, moonshine concepts, Rademacher sums and their significance in the quantum world. You should not miss their eccentric proposal of relating Bell's theorem to the Baby Monster group. Their hyperbolic polygons show a considerable singularity/cusp structure that our modern age of computers is able to capture. Henri Poincar{\'e} would have been happy to see it.Comment: new version expanded for publication in Mathematics (MDPI), special issue "Mathematical physics" initial: Trick or Truth: the Mysterious Connection Between Physics and Mathematics, FQXi essay contest - Spring, 201

The graph distance game

In the graph distance game, two players alternate in constructing a maximal path. The objective function is the distance between the two endpoints of the path, which one player tries to maximize and the other tries to minimize. In this note, we examine the distance game for various graphs, and provide general bounds, exact results for special graphs, and an algorithm for trees. Computer calculations suggest interesting conjectures for grids

On decay centrality in graphs

The decay centrality of a vertex v in a graph G with respect to a parameter d¿(0,1) is a polynomial in d such that for fixed k the coefficient of dk is equal to the number of vertices of G at distance k from v. This invariant (introduced independently by Jackson and Wolinsky in 1996 and Dangalchev in 2011) is considered as a replacement for the closeness centrality for graphs, however its unstability was pointed out by Yang and Zhuhadar in 2011. We explore mathematical properties of decay centrality depending on the choice of parameter d and the stability of vertex ranking based on this centrality index.Peer ReviewedPreprin