Let q be a prime a power and k an integer such that 3 ≤ k ≤ q. In this paper we present a method using Latin squares to construct adjacency matrices of k-regular bipartite graphs of girth 8 on 2(kq − q) vertices. Some of these graphs have the smallest number of vertices among the known regular graphs with girth 8.

1. Any vertex that is incident to an observed edge is observed. 2. Any edge joining two observed vertices is observed. The power domination problem is a variant of the classical domination problem in graphs and is defined as follows. Given an undirected graph G = (V, E), the problem is to find a minimum vertex set SP ⊆ V , called the power dominating set of G, such that all vertices in G are ob...

I have three goals for this lecture. The first is to introduce one of the most important familes of graphs: expander graphs. They are the source of much combinatorial power, and the counterexample to numerous conjectures. We will become acquainted with these graphs by examining random walks on them. To facilitate the analysis of random walks, we will examine these graphs through their adjacency...

We present a novel algorithm that generates scale free small world graphs such as those found in the World Wide Web,social and metabolic networks. We use the generated graphs to study the dynamics of a realistic search strategy on the graphs, and find that they can be navigated in a very short number of steps. Small world and scale free graphs, which are at the heart of systems as diverse as th...

It has been shown that there is a Hamilton cycle in every connected Cayley graph on any group G whose commutator subgroup is cyclic of prime-power order. This note considers connected, vertex-transitive graphs X of order at least 3, such that the automorphism group of X contains a vertex-transitive subgroup G whose commutator subgroup is cyclic of prime-power order. We show that of these graphs...

A hierarchical approach to the decomposition of graphs is introduced which is related to the notion of tree decomposition. On this basis a hierarchical automaton model for graphs is deened. We show that this automaton model is (relative to an appropriate class of graphs) equivalent to monadic second-order logic in expressive power, properly strengthening previous results on monadic second-order...

let $n$ be any positive integer, the friendship graph $f_n$ consists of $n$ edge-disjoint triangles that all of them meeting in one vertex. a graph $g$ is called cospectral with a graph $h$ if their adjacency matrices have the same eigenvalues. recently in href{http://arxiv.org/pdf/1310.6529v1.pdf}{http://arxiv.org/pdf/1310.6529v1.pdf} it is proved that if $g$ is any graph cospectral with $f_n$...

Due to the limited resources available in the wireless ad hoc networking nodes, the scalability is crucial for network operations. One effective approach is to maintain only a sparse spanner of a linear number of links while still preseving the power-efficient route for any pair of nodes. For any spanner G, its power stretch factor is defined as the maximum ratio of the minimum power needed to ...

The network alignment problem asks for the best correspondence between two given graphs, so that the largest possible number of edges are matched. This problem appears in many scientific problems (like the study of protein-protein interactions) and it is very closely related to the quadratic assignment problem which has graph isomorphism, traveling salesman and minimum bisection problems as par...

Let q be a prime power and g ∈ {6, 8, 12}. In this paper we obtain (q, g)-graphs on 2qg/2−3(q2 − 1) vertices for g = 6, 8, 12 as subgraphs of known (q + 1, g)-cages. We also obtain (q − r, 6)-graphs on 2(q2 − rq − 1) vertices, and (q − r, 8)-graphs on 2(q3−rq2−q+r) vertices where r is a positive integer such that q ≥ r+3. Furthermore, we find (q − 1, 12)-graphs of order 2q2(q3 − q2 − q + 1). So...