Let GF(q) denote the finite field with q elements. The Paley graph Paley(q) is defined to be a graph with vertex set GF(q) such that two vertices a and b are joined with an edge if a-b is a non-zero square. If we assume q ≡ 1(mod4) , then this graph is undirected. In this paper, our aim is to compute the topological indices of Paley(q) such as the Wiener, PI and Szeged indices of this graph.

Cop Robber game is a two player game played on an undirected graph. In this game, the cops try to capture a robber moving on the vertices of the graph. The cop number of a graph is the least number of cops needed to guarantee that the robber will be caught. We study cop-edge critical graphs, i. e. graphs G such that for any edge e in E(G) either c(G-e) < c(G) or c(G-e) > c(G). In this article, we study the edge criticality of generalized Petersen graphs and Paley graphs.

In this paper we first consider and study certain edge-transitive connected graphs, such as the Hamming graphs, the Paley graphs and the Boolean lattice. Then as a consequence, we obtain the Wiener and the hyper-Wiener indices of these graphs.

Please click on PDF to view the abstract

The study of classical Littlewood-Paley operators has an intrinsic interest for their essential role in harmonic analysis due to their applications in PDEs and other fields. One of the Littlewood-Paley operators is g λ *  operator which its p,p  strong boundedness depends on the parameter λ . For example, Fefferman showed strong boundedness of classical g λ *  for 1max 1, 2 p .  In this work, We consider the Laplace-Bessel differential operator and correspondingly we define the relevant Littlewood-Paley operator g B,λ *  to investigate both L p,ν - boundedness of g B,λ *  for 2≤P<∞  and λ>1+ 2v n and its unboundedness for 0

We determine the forbidden induced subgraphs for the intersection of the classes of chordal bipartite graphs and line graphs of acyclic directed graphs. This is a first step towards finding the forbidden induced subgraphs for the class of line graphs of directed graphs.

A set W ,V (G) is called a resolving set, if for every two distinct vertices u,v 2 V (G) there exists w 2 W such that d(u, w) 6= d(v, w), where d(x,y) is the distance between the vertices x and y. A resolving set for G with minimum cardinality is called a metric basis. A graph with a unique metric basis is called a uniquely dimensional graph. In this paper, we establish a family of graph called Solis graph, and we prove that if G is a minimal edge unique base graph with the base of size two, then G belongs to the Solis graphs family. Finally, an algorithm is given for , nding the metric dimension of a Solis graph.

The Hosoya index $Z(G)$ of a graph $G$ is the total number of matchings in it. In this paper, the recursive formulas of the Hosoya index of semitotal graph $Q(G)$ and total graph $T(G)$ for certain graphs $G$ are obtained. Moreover, we obtain the bounds of the Hosoya index of semitotal and total graphs of a connected graph $G$.