Unlike an ordinary fuzzy set, the concept of intuitionistic fuzzy set (IFS), characterized both by a membership degree and by a non-membership degree, is a more flexible way to capture uncertainty. In this paper, we have classified the states of intuitionistic Markov chain (IMC) [1] and analyzed the long-run behavior of the system.

A grid is a large-scale geographically distributed hardware and software infrastructure composed of heterogeneous networked resources owned and shared by multiple administrative organizations which are coordinated to provide transparent, dependable, pervasive and consistent computing support to a wide range of applications. One of the major problems in graph theory is to find the oriented diameter of a graph $G$, which is defined as the smallest diameter among the diameter of all strongly connected orientations. The problem is proved to be NP-complete. In this paper, we obtain the oriented diameter of grids.

By a \((1,1)\) edge-magic labeling of a graph \( G(V, E) \), we mean a bijection \( f \) from \( V \cup E \) to \(\{1, \dots, |V| + |E|\}\) such that for all edges \( uv \in E(G) \), the value \( f(u) + f(v) + f(uv) \) is constant. We provide a different proof of a well-known result in additive number theory by Paul Erdős and, interestingly, demonstrate a practical application of this result. Additionally, we make some progress using computational methods towards the conjecture proposed by Yegnanarayanan: “Every graph on \( p \geq 9 \) vertices can be embedded as a subgraph of some \((1,1)\) edge-magic graph.”

In this paper, an \( n \times n \) fully fuzzy linear system is solved by decomposing the positive definite symmetric coefficient matrix using trapezoidal fuzzy number matrices through Cholesky and LDLT decomposition methods. The effectiveness of these methods is illustrated with a numerical example.

Given an undirected 2-edge connected graph, finding a minimum 2-edge connected spanning subgraph is NP-hard. We solve the problem for Butterfly network, Benes network, Honeycomb network and Sierpiński gasket graph.

The Terminal Wiener index \( TW(G) \) of a graph \( G \) is defined as the sum of the distances between all pairs of pendant vertices. In this paper, we derive an explicit formula for calculating the Terminal Wiener index for Detour-saturated trees and Nanostar Dendrimers.

Let \( G = (V, E) \) be a graph. A set \( W \subset V \) of vertices **resolves** \( G \) if every vertex of \( G \) is uniquely determined by its vector of distances to the vertices in \( W \). The **metric dimension** of \( G \) is the minimum cardinality of a resolving set.

By imposing conditions on \( W \), we obtain **conditional resolving sets**.

A proper vertex coloring (no two adjacent vertices have the same color) of a graph \( G \) is said to be acyclic if the induced subgraph of any two color classes is acyclic. The minimum number of colors required for an acyclic coloring of a graph \( G \) is called its acyclic chromatic number and is denoted by \( a(G) \). In this paper, we determine the exact value of the acyclic chromatic number for the central and total graphs of the path \( P_n \), and the Fan graph \( F_{m,n} \).

Eigenvalues of a graph are the eigenvalues of its adjacency matrix. The multiset of eigenvalues is called the \emph{spectrum}. The energy of a graph is defined as the sum of the absolute values of its eigenvalues. In this paper, we devise an algorithm that generates the adjacency matrix of \( WK \)-recursive structures \( WK(3, L) \) and \( WK(4, L) \), and use it to effectively compute the spectrum and energy of these graphs.

Given a connected \((p, q)\) graph \(G\) with a number of central vertices, we form a new graph \(G^*\) as follows:

– **Vertex Set**: \(V(G^*) = V(G)\).
– **Edge Set**: Delete all the edges of \(G\). Introduce an edge between every central vertex and each non-central vertex of \(G\). Allow every pair of central vertices to be adjacent.

In this paper, we probed \(G^*\) and deduced a number of results.