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.

Structures realized by arrangements of regular hexagons in the plane are of interest in the chemistry of benzenoid hydrocarbons, where perfect matchings correspond to Kekulé structures which feature in the calculation of molecular energies associated with benzenoid hydrocarbon molecules. Mathematically, assembling in predictable patterns is equivalent to packing in graphs. An \( H \)-packing of a graph \( G \) is a set of vertex-disjoint subgraphs of \( G \), each of which is isomorphic to a fixed graph \( H \). If \( H \) is the complete graph \( K_2 \), the maximum \( H \)-packing problem becomes the familiar maximum matching problem. In this paper, we find an \( H \)-packing of an armchair carbon nanotube with \( H \) isomorphic to \( P_4 \), \emph{1, 4-dimethyl cyclohexane}, and \( C_6 \). Further, we determine the \( H \)-packing of a zigzag carbon nanotube with \( H \) isomorphic to \emph{1, 4-dimethyl cyclohexane}.

ll graphs considered in our study are simple, finite and undirected. A graph is equitable total domination edge addition critical (stable) if the addition of any arbitrary edge changes (does not change) the equitable total domination number. In this paper, we introduce the following new parameters: equitable independent dom- ination number, equitable total domination number and equitable connected domination number and study their stability upon edge addition, on special families of graphs namely cycles, paths and com- plete bipartite graphs. Also the relation among the above parameters is established.

Let \(G(V, E)\) be a simple graph. For a labeling \(\partial: V \cup E \to \{1, 2, 3, \dots, k\}\), the weight of a vertex \(x\) is defined as \(wt(x) = \partial(x) + \sum\limits_{xy \in E} \partial(xy)\). \(\partial\) is called a vertex irregular total \(k\)-labeling if for every pair of distinct vertices \(x\) and \(y\), \(wt(x) \neq wt(y)\). The minimum \(k\) for which the graph \(G\) has a vertex irregular total \(k\)-labeling is called the total vertex irregularity strength of \(G\) and is denoted by \(tvs(G)\). In this paper, we obtain a bound for the total vertex irregularity strength of honeycomb and honeycomb derived networks.

A linear layout, or simply a layout, of an undirected graph \( G = (V, E) \) with \( n = |V| \) vertices is a bijective function \( \phi: V \to \{1, 2, \dots, n\} \). A \( k \)-coloring of a graph \( G = (V, E) \) is a mapping \( \kappa: V \to \{c_1, c_2, \dots, c_k\} \) such that no two adjacent vertices have the same color. A graph with a \( k \)-coloring is called a \( k \)-colored graph.

A colored layout of a \( k \)-colored graph \( (G, \kappa) \) is a layout \( \phi \) of \( G \) such that for any \( u, x, v \in V \), if \( (u, v) \in E \) and \( \phi(u) < \phi(x) < \phi(v) \), then \( \kappa(u) \neq \kappa(x) \). Given a \( k \)-colored graph \( (G, \kappa) \), the problem of deciding whether there is a colored layout \( \phi \) of \( (G, \kappa) \) is NP-complete. In this paper, we introduce the concept of chromatic layout of \( G \) and determine the chromatic layout number for paths and cycles.

Let \( G(V, E) \) be a simple graph. For a labeling \( \partial: V \cup E \to \{1, 2, 3, \dots, k\} \), the weight of a vertex \( x \) is defined as

\[
wt(x) = \partial(x) + \sum_{xy \in E} \partial(xy).
\]

The labeling \( \partial \) is called a vertex irregular total \( k \)-labeling if for every pair of distinct vertices \( x \) and \( y \), \( wt(x) \neq wt(y) \). The minimum \( k \) for which the graph \( G \) has a vertex irregular total \( k \)-labeling is called the total vertex irregularity strength of \( G \) and is denoted by \( tvs(G) \). In this paper, we obtain a bound for the total vertex irregularity strength of honeycomb and honeycomb derived networks.

Graph embedding is an important technique used in the study of computational capabilities of processor interconnection networks and task distribution. In this paper, we present an algorithm for embedding the Hypercubes into Banana Trees and Extended Banana Trees and prove its correctness using the Congestion lemma and Partition lemma.

A kernel in a directed graph \(D(V, E)\) is a set \(S\) of vertices of \(D\) such that no two vertices in \(S\) are adjacent and for every vertex \(u\) in \(V \setminus S\), there is a vertex \(v\) in \(S\) such that \((u, v)\) is an arc of \(D\). The problem of existence of a kernel is NP-complete for a general digraph. In this paper, we introduce the acyclic kernel problem of an undirected graph \(G\) and solve it in polynomial time for uniform theta graphs and even quasi-uniform theta graphs.

Given a graph \( G = (V, E) \), a labeling \( \partial: V \cup E \to \{1, 2, \dots, k\} \) is called an edge irregular total \( k \)-labeling if for every pair of distinct edges \( uv \) and \( xy \), \( \partial(u) + \partial(uv) + \partial(v) \neq \partial(x) + \partial(xy) + \partial(y) \). The minimum \( k \) for which \( G \) has an edge irregular total \( k \)-labeling is called the total edge irregularity strength of \( G \). In this paper, we examine the hexagonal network, which is a well-known interconnection network, and obtain its total edge irregularity strength.

Graph embedding problems have gained importance in the field of interconnection networks for parallel computer architectures. In this paper, we prove that grid and cylinder are the subgraphs of certain circulant networks. Further, we present an algorithm to embed tori into certain circulant networks with dilation\(2\) and vice-versa.

Broadcasting is a fundamental information dissemination problem in a connected graph, in which one vertex called the originator disseminates one or more messages to all other vertices in the graph. \(A\)-broadcasting is a variant of broadcasting in which an informed vertex can disseminate a message to at most \(k\) uninformed vertices in one unit of time. In general, solving the broadcast problem in an arbitrary graph is NP-complete. In this paper, we obtain the \(k\)-broadcast time of the Sierpiński gasket graphs for all \(k \geq 1\).

Let \( G = (V, E) \) be a graph with vertex set \( V \) and edge set \( E \). Let \( \text{diam}(G) \) denote the diameter of \( G \) and \( d(u, v) \) denote the distance between the vertices \( u \) and \( v \) in \( G \). An antipodal labeling of \( G \) with diameter \( d \) is a function \( f \) that assigns to each vertex \( u \), a positive integer \( f(u) \), such that \( d(u, v) + |f(u) – f(v)| \geq d \), for all \( u, v \in V \). The span of an antipodal labeling \( f \) is \( \max \{|f(u) – f(v)| : u, v \in V(G)\} \). The antipodal number for \( G \), denoted by \( \text{an}(G) \), is the minimum span of all antipodal labelings of \( G \). Determining the antipodal number of a graph \( G \) is an NP-complete problem. In this paper, we determine the antipodal number of certain graphs.

The concept of fuzzy local \(\omega\)-language and Büchi fuzzy local \(\omega\)-language are defined in \([1,2]\). In this paper, we define Landweber fuzzy local \(\omega\)-language and study their closure properties and also give an automata characterization for it. Finally, we conclude the hierarchy among the subclasses of fuzzy regular \(\omega\)-languages.

In this paper, we have calculated the combinatorial counting relations varying over the \(3\)-vertex paths of a simple graph \(G\), by restricting our attention to \(C_3\), \(C_4\)-free graphs.

A kernel in a directed graph \(D(V, E)\) is a set \(S\) of vertices of \(D\) such that no two vertices in \(S\) are adjacent and for every vertex \(u\) in \(V \setminus S\) there is a vertex \(v\) in \(S\) such that \((u, v)\) is an arc of \(D\). The problem of existence of a kernel itself is NP-complete for a general digraph. But in this paper, we solve the strong kernel problem of certain oriented networks in polynomial time.

A double shell is defined to be two edge-disjoint shells with a common apex. In this paper, we prove that double shells (where the shell orders are \(m\) and \(2m+1\)) with exactly two pendant edges at the apex are \(k\)-graceful when \(k=2\). We extend this result to double shells of any order \(m\) and \(\ell\) (where \(m \geq 3\) and \(\ell \geq 3\)) with exactly two pendant edges at the apex.

A book consists of a line in the 3-dimensional space, called the spine, and a number of pages, each a half-plane with the spine as boundary. A book embedding \((\pi, p)\) of a graph consists of a linear ordering \(\pi\) of vertices, called the spine ordering, along the spine of a book and an assignment \(p\) of edges to pages so that edges assigned to the same page can be drawn on that page without crossing. That is, we cannot find vertices \(u, v, x, y\) with \(\pi(u) < \pi(x) < \pi(v) 2\) and \(C_n\) are given. If \(G\) is any graph, an upper bound for the page number of the Mycielski of \(G\) is given. When \(G\) and \(H\) are any two graphs with page number \(k\) and \(l\), it is proved that the amalgamation of \(G\) and \(H\) can be embedded in a \max(k, l)\) pages. Further, we remark that the amalgamation of \(G\) with itself requires the same number of pages as \(G\), irrespective of the vertices identified in the two copies of \(G\), to form an amalgamation.

In 2004, Blinco et al [1] introduced \(\gamma\)-labeling. A function \(h\) defined on the vertex set of a graph \(G\) with \(n\) edges is called a \(\gamma\)-labeling if:

1. \(h\) is a \(p\)-labeling of \(G\),
2. \(G\) admits a tripartition \((A, B, C)\) with \(C = \{c\}\) and there exists \(\overline{b} \in B\) such that \((\overline{b}, c)\) is the unique edge with the property that \(h(c) – h(\overline{b}) = n\),
3. for every edge \(\{a, v\} \in E(G)\) with \(a \in A\), \(h(a) < h(v)\).

In [1] they have also proved a significant result on graph decomposition that if a graph \(G\) with \(n\) edges admits a \(\gamma\)-labeling then the complete graph \(K_{2cn+1}\) can be cyclically decomposed into \(2cn + 1\) copies of the graph \(G\), where \(c\) is any positive integer.

Motivated by the result of Blinco et al [1], in this paper, we prove that the well-known almost-bipartite graph, the grid with an additional edge, \((P_m \Box P_n) + \hat{e}\), admits \(\gamma\)-labeling. Further, we discuss a related open problem.

A set \( S \) of vertices of a graph \( G(V,E) \) is a \emph{dominating set} if every vertex of \( V \setminus S \) is adjacent to some vertex in \( S \). A dominating set is said to be \emph{efficient} if every vertex of \( V \setminus S \) is dominated by exactly one vertex of \( S \). A paired-dominating set is a dominating set whose induced subgraph contains at least one perfect matching. A set \( S \) of vertices in \( G \) is a total dominating set of \( G \) if every vertex of \( V \) is adjacent to some vertex in \( S \). In this paper, we construct a minimum paired dominating set and a minimum total dominating set for the infinite diamond lattice. The total domatic number of \( G \) is the size of a maximum cardinality partition of \( V \) into total dominating sets. We also demonstrate that the total domatic number of the infinite diamond lattice is 4.

In this paper we introduce right angle path and layer of an array. We construct Kolakoski array and study some combinatorial proper-ties of Kolakoski array. Also we obtain recurrence relation for layers and special elements.

An \emph{eternal 1-secure} set of a graph \(G = (V, E)\) is defined as a set \(S_0 \subseteq V\) that can defend against any sequence of single-vertex attacks by means of single guard shifts along edges of \(G\). That is, for any \(k\) and any sequence \(v_1, v_2, \ldots, v_k\) of vertices, there exists a sequence of guards \(u_1, u_2, \ldots, u_k\) with \(u_i \in S_{i-1}\) and either \(u_i = v_i\) or \(u_iv_i \in E\), such that each set \(S_i = (S_{i-1} -\{u_i\}) \cup \{v_i\}\) is dominating. It follows that each \(S_i\) can be chosen to be an eternal 1-secure set. The \emph{eternal 1-security number}, denoted by \(\sigma_1(G)\), is defined as the minimum cardinality of an eternal 1-secure set. This parameter was introduced by Burger et al. [3] using the notation \(\gamma_\infty\). The \emph{eternal \(m\)-security} number \(\sigma_m(G)\) is defined as the minimum number of guards to handle an arbitrary sequence of single attacks using multiple-guard shifts. A suitable placement of the guards is called an \emph{eternal \(m\)-secure} set. It was observed that \(\gamma(G) \leq \sigma_m(G) \leq \beta(G)\). In this paper, we obtain specific values of \(\sigma_m(G)\) for certain classes of graphs, namely circulant graphs, generalized Petersen graphs, binary trees, and caterpillars.