Eigenvalues of a graph are the eigenvalues of its adjacency matrix. The multiset of eigenvalues is called the \({spectrum}\). 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 with a number of central vertices, form a new graph \(G^*\) as follows: \(V(G^*) = V(G)\); Delete all the edges of \(G\). Introduce an edge between every central vertex to each and every 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.

The \(k\)-rainbow domination is a variant of the classical domination problem in graphs and is defined as follows: Given an undirected graph \(G = (V, E)\) and a set of \(k\) colors numbered \(1, 2, \dots, k\), we assign an arbitrary subset of these colors to each vertex of \(G\). If a vertex is assigned the empty set, then the union of color sets of its neighbors must be \(k\) colors. This assignment is called the \(k\)-rainbow dominating function of \(G\). The minimum sum of numbers of assigned colors over all vertices of \(G\), is called the \(k\)-rainbow domination number of \(G\). In this paper, we present some bounds on the \(3\)-rainbow domination number of circulant networks and grid 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.