An incidence of a graph $G$ is a pair $(x,e)$, where $x$ is a vertex of $G$ and $e$ is an edge of $G$ incident to $x$. Two incidences $(x,e)$ and $(y,f)$ are adjacent if any one of the following holds: (i) $x=y$, or (ii) $e=f$, or (iii) $xy=e$ or $f$. In [3], Brualdi and Massey introduced the concept of incidence coloring of a graph $G$ as a mapping from the set of incidences of $G$ to a finite set of colors such that adjacent incidences receive distinct colors. A signed graph $(G,\sigma)$ consists of a graph $G$ and the signature $\sigma : E(G)\rightarrow \{+1,-1\}$. In this paper, we define an incidence coloring of signed graphs as a natural generalization of the usual notion of incidence coloring of unsigned graphs. We prove that our definition is compatible with switching operation. We also prove that the incidence chromatic number (in signed sense) of a signed graph $(G,\sigma)$ coincide with the incidence chromatic number (in the usual unsigned sense) of its underlying graph $G$. The exact value or upper bounds which are known for the incidence chromatic numbers of some well-known families of unsigned graphs are also mentioned for their signed versions, namely, signed cycles, signed trees, signed complete graphs and signed toroidal grids.

Let denote the minimum cardinality among all maximal independent sets of containing . Then is called the independent dom-saturation number of . In this paper, we study the effect of removal of an edge on the independent dom-saturation number of a graph. We also study the concept of -subdivision number of a graph and initiate a study of edge independent dom-saturation number of a graph

A triangulation of a graph , denoted by , is obtained from by replacing each edge in by a complete graph . In this study, the Laplacian polynomial of is investigated. Moreover, an explicit formula for the number of spanning trees of is determined based on the analysis of its Laplacian polynomial properties. Two complex networks and obtained by iterated triangulations executed to cycle graph are studied. Furthermore, we obtain exact formulas for the number of spanning trees of these complex networks. The validity of the proposed formulas is numerically tested by Matlab.

The objective of the study is to present an overview of network analysis with a special focus on the construction and application of complete graphs in organizational networks. Network analysis is approached both statistically and algorithmically to understand the relationships among system components. In this context, complete graphs allow for the effective implementation of various algorithms to solve the critical path, minimum spanning tree (MST), maximal flow and shortest route problems. These problems are addressed using different classical graph theory techniques such as the PERT (Program Evaluation and Review Technique) method to solve the critical path, Prim’s algorithm for finding the MST, the Ford-Fulkerson algorithm for computing the maximum flow in a network, and the traveling salesman algorithm for determining the shortest route of a complete graph with n vertices in a single network design with an appropriate numerical example. The comprehensive connectivity in complete graphs ensures greater flexibility and accuracy in solving complex network-related problems in organizational and operational settings.

Regarding the context of circuit representation theory of Markov processes, we study the alternative unique representation by directed circuits and weights of the corresponding non-reversible Markov chain describing a random walk with steps (+3) or (–2), as an application of the existence of non-reversible circuit chains.

In this paper, we develop an optimization method for solving the problem of the curse of dimensionality in dynamic programming. The method has the advantage of providing a cost for decision-making and eliminates the curse of dimensionality, which restricts the application of dynamic programming to only small classes of problems and, therefore, restricts the real-life application of dynamic programming. We established some relationships between geometric programming and dynamic programming parameters. We applied the method to a problem on course allocation and obtained the cost for course allocation and the optimal decision policy to be (0.1, 0.2, 1, 3).

A subset of is a neighborhood of if can be represented as the union of the induced subgraphs of the closed neighborhoods of each vertex in . Moreover, the neighborhood number denoted by is the minimum cardinality of the neighborhood set of . The neighborhood polynomial of a graph of order is defined as . In this paper, we obtain the neighborhood polynomial of .

For a nontrivial connected graph , a nonempty subset is a total offensive alliance in if is an offensive alliance and every vertex in has at least one neighbor within . The minimum cardinality of a total offensive alliance in is called the total offensive alliance number of , denoted by . In this paper, we investigate the total offensive alliance of total graphs generated from graphs with maximum degree 2 and present the characterization of total offensive alliance and their corresponding total offensive alliance number.

We investigate the algebraic structures of square grid graphs and obtain isomorphism theorems.

Let be a connected graph. Then a non-empty is a 2 movable dominating set of if is a dominating set and for every pair is a dominating set in , or there exist such that and are adjacent to and , respectively, and is a dominating set in . The 2-movable domination number of , denoted by , is the minimum cardinality of a 2-movable dominating set of G. A 2-movable dominating set with cardinality equal to is called -set of .