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.

The distance $d(u,v)$ between vertices $u$ and $v$ of a connected graph $G$ is equal to the number of edges in a minimal path connecting them. The transmission of a vertex $v$ is defined by $\sigma(v)=\sum\limits_{u\in V(G)}{d(v,u)}$. A topological index is said to be a transmission-based topological index (TT index) if it includes the transmissions $\sigma(u)$ of vertices of $G$. Because $\sigma(u)$ can be derived from the distance matrix of $G$, it follows that transmission-based topological indices form a subset of distance-based topological indices.In this article we survey some results on the computation of some transmission-based graph invariants of intersection graph, hypercube graph, Kneser graph, unitary Cayley graph and Paley 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.

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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

The independence graph Ind(G) of a graph G is the graph with vertices as maximum independent sets of G and two vertices are adjacent, if and only if the corresponding maximum independent sets are disjoint. In this work, we find the independence graph of Cartesian product of d copies of complete graphs Kq, which is known as the Hamming graph H(d, q). Greenwell and Lovasz [7] found that the independence number of direct product of d copies of Kq as qd−1. We prove that the independence number of Hamming graph H(d, q), which is cartesian product of d copies of Kq, is also qd−1. As an application of our findings, we find answers for rook problem in higher dimensional square chess board.

In this paper, we find the star chromatic number of central graph of complete bipartite graph and corona graph of complete graph with path and cycle.

Let R be a commutative ring and M be an R-module. The M-intersection graph of ideals of R, denoted by GM(R) is a graph with the vertex set I(R) ∗, , and two distinct vertices I and J are adjacent if and only if IM ∩,JM ̸, = 0. In this paper, we study GR/J (R/I), where I and J are ideals of R and I ⊆,J. We characterize all ideals I and J for which GR/J (R/I) is planar, outerplanar or ring graph.