Représentations sphériques des groupes agissant transitivement sur un arbre semi-homogène
For each integer and each finite graph , we construct a Coxeter group and a non positively curved polygonal complex on which acts properly cocompactly, such that each polygon of has edges, and the link of each vertex of is isomorphic to . If is a “generalized -gon”, then is a Tits building modelled on a reflection group of the hyperbolic plane. We give a condition on for to be non enumerable (which is satisfied if is a thick classical generalized -gon). On the other hand,...
For an ordered set of vertices and a vertex in a connected graph , the (metric) representation of with respect to is the -vector , where represents the distance between the vertices and . The set is a resolving set for if distinct vertices of have distinct representations with respect to . A resolving set of minimum cardinality is called a minimum resolving set or a basis and the cardinality of a basis for is its dimension . A set of vertices in is a dominating set...
A directed Cayley graph is specified by a group and an identity-free generating set for this group. Vertices of are elements of and there is a directed edge from the vertex to the vertex in if and only if there is a generator such that . We study graphs for the direct product of two cyclic groups and , and the generating set . We present resolving sets which yield upper bounds on the metric dimension of these graphs for and .
We analyse the spectral phase diagram of Schrödinger operators on regular tree graphs, with the graph adjacency operator and a random potential given by random variables. The main result is a criterion for the emergence of absolutely continuous spectrum due to fluctuation-enabled resonances between distant sites. Using it we prove that for unbounded random potentials spectrum appears at arbitrarily weak disorder in an energy regime which extends beyond the spectrum of. Incorporating...
Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and to a vertex in V-S. The restrained domination number of G, denoted by , is the minimum cardinality of a restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We show that if U is a unicyclic graph of order n, then , and provide a characterization of graphs achieving this bound.
For any nontrivial connected graph and any graph , the -degree of a vertex in is the number of copies of in containing . is called -continuous if and only if the -degrees of any two adjacent vertices in differ by at most 1; is -regular if the -degrees of all vertices in are the same. This paper classifies all -continuous graphs with girth greater than 3. We show that for any nontrivial connected graph other than the star , , there exists a regular graph that is not...
This paper analyzes the proof-theoretic strength of an infinite version of several theorems from algorithmic graph theory. In particular, theorems on reachability matrices, shortest path matrices, topological sorting, and minimal spanning trees are considered.