In this paper we compute the \(P_3\)-forcing number of honeycomb network. A dynamic coloring of the vertices of a graph \(G\) starts with an initial subset \(S\) of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set \(S\) is called a forcing set of \(G\) if, by iteratively applying the forcing process, every vertex in G becomes colored. If the initial set \(S\) has the added property that it induces a subgraph of \(G\) whose components are all paths of length 3, then \(S\) is called a \(P_3\)-forcing set of \(G\). A Ps-forcing set of \(G\) of minimum cardinality is called the \(P_3\)-forcing number of G denoted by \(ZP_3(G)\).

In this paper, we introduced a new concept called nonsplit monophonic set and its relative parameter nonsplit monophonic number \(m_{ns}(G)\). Some certain properties of nonsplit monophonic sets are discussed. The nonsplit monophonic number of standard graphs are investigated. Some existence theorems on nonsplit monophonic number are established.

In graph theory and network analysis, centrality measures identify the most important vertices within a graph. In a connected graph, closeness centrality of a node is a measure of centrality, calculated as the reciprocal of the sum of the lengths of the shortest paths between the node and all other nodes in the graph. In this paper, we compute closeness centrality for a class of neural networks and the sibling trees, classified as a family of interconnection networks.

Let \(G = (V, E)\) be a graph. A total dominating set of \(G\) which intersects every minimum total dominating set in \(G\) is called a transversal total dominating set. The minimum cardinality of a transversal total dominating set is called the transversal total domination number of G, denoted by \(\gamma_{tt}(G)\). In this paper, we begin to study this parameter. We calculate \(\gamma_{tt}(G)\) for some families of graphs. Further some bounds and relations with other domination parameters are obtained for \(\gamma_{tt}(G)\).

Let \(G\) be any graph. The concept of paired domination was introduced having gaurd backup concept in mind. We introduce pendant domination concept, for which at least one guard is assigned a backup, A dominating set \(S\) in \(G\) is called a pendant dominating set if \((S)\) contains at least one pendant vertex. The least cardinality of a pendant dominating set is called the pendant domination number of G denoted by \(\gamma_{pe}(G)\). In this article, we initiate the study of this parameter. The exact value of \(\gamma_{pe}(G)\) for some families of standard graphs are obtained and some bounds are estimated. We also study the proprties of the parameter and interrelation with other invarients.

Transportation Problem (TP) is the exceptional case to obtain the minimum cost. A new hypothesis is discussed for getting minimal cost in transportation problem in this paper and also Vogel’s Approximation Method (VAM) and MODI method are analyzed with the proposed method. This approach is examined with various numerical illustrations.

This paper mainly surveys the literature on bulk queueing models and its applications. Distributed Different systems in the zone of queuing speculation merging mass queuing architecture. These mass queueing models are often related to confirm the clog issues. Through this diagram, associate degree challenge has been created to envision the paintings accomplished on mass strains, showing various wonders and also the goal is to present enough facts to inspectors, directors and enterprise those that are dependent on the usage of queueing hypothesis to counsel blockage troubles and need to find the needs of enthusiasm of applicable models near the appliance.

Frank Harary and Allen J. Schwenk have given a formula for counting the number of non-isomorphic caterpillars on \(n\) vertices with \(n ≥ 3\). Inspired by the formula of Frank Harary and Allen
J. Schwenk, in this paper, we give a formula for counting the number of non-isomorphic caterpillars with the same degree sequence.

The line graph \(L(G)\) of a connected graph G, has vertex set identical with the set of edges of \(G\), and two vertices of \(L(G)\) are adjacent if and only if the corresponding edges are adjacent in \(G\). Ivan Gutman et al examined the dependency of certain physio-chemical properties of alkanes in boiling point, molar volume, and molar refraction, heat of vapourization, critical temperature, critical pressure and surface tension on the Bertz indices of \(L'(G)\) Dobrynin and Melnikov conjectured that there exists no nontrivial tree \(T\) and \(i≥3\), such that \(W(L'(T)) = W(T)\). In this paper we study Wiener and Zagreb indices for line graphs of Complete graph, Complete bipartite graph and wheel graph.

A set S of vertices in a graph G is called a dominating set of G if every vertex in V(G)\S is adjacent to some vertex in S. A set S is said to be a power dominating set of G if every vertex in the system is monitored by the set S following a set of rules for power system monitoring. The power domination number of G is the minimum cardinality of a power dominating set of G. In this paper, we solve the power domination number for certain nanotori such as H-Naphtelanic, \(C_5C_6C_7[m,n]\) nanotori and \(C_4C_6C_8[m,n]\) nanotori.