A digraph G is finite and is denoted as \(G(V,E)\) with \(V\) as set of nodes and E as set of directed arcs which is exact. If \((u, v)\) is an arc in a digraph \(D\), we say vertex u dominates vertex v. A special digraph arises in round robin tournaments. Tournaments with a special quality \(Q(n, k)\) have been studied by Ananchuen and Caccetta. Graham and Spencer defined tournament with \(q\) vertices
where \(q \equiv 3 (mod 4)\) is a prime. It was named suitably as Paley digraphs in respect deceased Raymond Paley, he was the person used quadratic residues to construct Hadamard matrices more than 50 years earlier. This article is based on a special class of graph called Paley digraph which admits odd edge graceful, super edge graceful and strong edge graceful labeling.

Molecular graphs are models of molecules in which atoms are represented by vertices and chemical bonds by edges of a graph. Graph invariant numbers reflecting certain structural features of a molecule that are derived from its molecular graph are known as topological indices. A topological index is a numerical descriptor of a molecule, based on a certain topological feature of the corresponding molecular graph. One of the most widely known topological descriptor is the Wiener index. The Weiner index \(w(G)\) of a graph G is defined as the half of the sum of the distances between every pair of vertices of \(G\). The construction and investigation of topological is one of the important directions in mathematical chemistry. The common neighborhood graph of G is denoted by con(\(G\)) has the same vertex set as G, and two vertices of con(\(G\)) are adjacent if they have a common neighbor in \(G\). In this paper we investigate the Wiener index of \(Y-tree,\, X-tree,\, con(Y-tree)\) and \(con(X-tree)\).

In the field of membrane computing. P system is a versatile model of computing, introduced by Paun [6], based on a combination of (i) the biological features of functioning of living cells and the members structure and (ii) the theoretical  concepts and results related to formal language theory. Among different Application areas of the model of P system, Ceterchi et al. [2] proposed an array-rewriting P system generating picture arrays based on the well-established notions in the area of array rewriting grammars and iso-array grammar have also been introduced. In this paper we consider structures in the two-dimensional plane called equi-triangular array composed of equilateral triangular array grammar and a corresponding P system, in the order to generate such structures. We Also examine the generative power of these new models of picture generation.

We disprove a conjecture proposed in [Gaspers et al., Discrete Applied Mathematics, 2010] and provide a new upper bound for the minimum number of brushes required to continually parallel clean a clique.

The strong chromatic index \( \chi’_s(G) \) of a graph \( G \) is the smallest integer \( k \) such that \( G \) has a proper edge \( k \)-coloring with the condition that any two edges at distance at most 2 receive distinct colors. It is known that \( \chi’_s(G) \leq 3\Delta – 2 \) for any \( K_4 \)-minor free graph \( G \) with \( \Delta \geq 3 \). We give a polynomial algorithm in order \( O(|E(G)|(n\Delta^2 + 2n + 14\Delta)) \) to strong color the edges of a \( K_4 \)-minor free graph with \( 3\Delta – 2 \) colors where \( \Delta \geq 3 \).

We introduce a new representation of MOFS of type \( F(m\lambda; \lambda) \), as a linear combination of \( \{0,1\} \) arrays. We use this representation to give an elementary proof of the classical upper bound, together with a structural constraint on a set of MOFS achieving the upper bound. We then use this representation to establish a maximality criterion for a set of MOFS of type \( F(m\lambda; \lambda) \) when \( m \) is even and \( \lambda \) is odd, which simplifies and extends a previous analysis \cite{ref3} of the case when \( m = 2 \) and \( \lambda \) is odd.

Arboricity is a graph parameter akin to chromatic number, in that it seeks to partition the vertices into the smallest number of sparse subgraphs. Where for the chromatic number we are partitioning the vertices into independent sets, for the arboricity we want to partition the vertices into cycle-free subsets (i.e., forests). Arboricity is NP-hard in general, and our focus is on the arboricity of cographs. For arboricity two, we obtain the complete list of minimal cograph obstructions. These minimal obstructions do generalize to higher arboricities; however, we no longer have a complete list, and in fact, the number of minimal cograph obstructions grows exponentially with arboricity.

We obtain bounds on their size and the height of their cotrees. More generally, we consider the following common generalization of colouring and partition into forests: given non-negative integers \( p \) and \( q \), we ask if a given cograph \( G \) admits a vertex partition into \( p \) forests and \( q \) independent sets. We give a polynomial-time dynamic programming algorithm for this problem. In fact, the algorithm solves a more general problem which also includes several other problems such as finding a maximum \( q \)-colourable subgraph, maximum subgraph of arboricity-\( p \), minimum feedback vertex set and the minimum weight of a \( q \)-colourable feedback vertex set.

Motivated by problems involving triangle-decompositions of graphs, we examine the facet structure of the cone \( \tau_n \) of weighted graphs on \( n \) vertices generated by triangles. Our results include enumeration of facets for small \( n \), a construction producing facets of \( \tau_{n+1} \) from facets of \( \tau_n \), and an arithmetic condition on entries of the normal vectors. We also point out that a copy of \( \tau_n \) essentially appears via the perimeter inequalities at one vertex of the metric polytope.

We consider the problem of detecting an intruder in a network where there are two types of detecting devices available. One device can determine the distance from itself to the intruder and the other can see the vertex it occupies as well as all adjacent vertices. The problem is to determine how many devices are required and where they should be placed in order to determine a single intruder’s location. We show that on the one hand, finding the minimum number of devices required to do this is easy when the network is a tree with at most one leaf adjacent to any vertex, while on the other hand finding this number is an NP-complete problem even for trees with at most two leaves adjacent to any vertex.

We present a 2-edge-coloured analogue of the duality theorem for transitive tournaments and directed paths. Given a 2-edge-coloured path \( P \) whose edges alternate blue and red, we construct a 2-edge-coloured graph \( D \) so that for any 2-edge-coloured graph \( G \),

\[
P \to G \iff G \not\to D.
\]

The duals are simple to construct, in particular \(|V(D)| = |V(P)| -1.\)