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In block encryption algorithms the only nonlinear component, which creates confusion and diffusion, is substitution box (S-Box). Therefore, carefully selection of S-Box is an important task. In this article, an algorithm is presented for the construction of 8 × 8 S-Box based on all subfields of Galois Field G F 2 16 and linear fractional transformation. Proposed algorithm is easy and simple but significantly complex for the attackers as it utilizes all subfields of G F (2 16) corresponding to all primitive irreducible polynomials for making G F (2 16) . An illustrating S-Box is also developed which is showing good structural characteristics such as it has no fixed point with 108.5 nonlinearity and 0.5020 strict avalanche. In the last, illustrated S-Box is applied on images for checking the performance under various criteria available in the literature. Obtained results of analyses are compared to well-known S-Boxes and found better in the comparison. [ABSTRACT FROM AUTHOR]

Metric-based parameters aided the study of the structural properties of graph network space like complexity, central tendency, connectivity, robustness, accessibility, and vulnerability. At the same time, these parameters played a pivotal role in rectifying problems in navigation, integer programming, optimization, pattern recognition and avoidance, combinatoric detection, backtracking in geographical routing, etc. In this paper, we considered the fractional edge dimension, a metric-based parameter in local environment by initiating the study of the fractional local edge dimensions of a graph (FLED). We define the aforesaid concept and give its bounds. We discussed the characterization problem for FLED equal to 1 , the realization problem, and its relation with the fractional metric dimensions. We also calculated the FLED of the Petersen graph, Coxeter graph, the families of cycle, complete, wheel, grid and complete r -partite graphs. [ABSTRACT FROM AUTHOR]

Let G = (V (G), E(G)) be a connected graph and x, y ∈ V (G), d(x, y) = min{ length of x − y path } and for e ∈ E(G), d(x, e) = min{d(x, a), d(x, b)}, where e = ab. A vertex x distinguishes two edges e1 and e2, if d(e1, x) ≠ d(e2, x). Let WE = {w1, w2, . . ., wk} be an ordered set in V (G) and let e ∈ E(G). The representation r(e | WE) of e with respect to WE is the k-tuple (d(e, w1), d(e, w2), . . ., d(e, wk)). If distinct edges of G have distinct representation with respect to WE, then WE is called an edge metric generator for G. An edge metric generator of minimum cardinality is an edge metric basis for G, and its cardinality is called edge metric dimension of G, denoted by edim(G). The circulant graph Cn(1, m) has vertex set {v1, v2, . . ., vn} and edge set {vivi+1 : 1 ≤ i ≤ n−1}∪{vnv1}∪{vivi+m : 1 ≤ i ≤ n−m}∪{vn−m+ivi : 1 ≤ i ≤ m}. In this paper, it is shown that the edge metric dimension of circulant graphs Cn(1, 2) and Cn(1, 3) is constant.

Apart from many branches in mathematical sciences, the main share of applications of graph theory nowadays is in chemistry. The graph-theoretical ideas have been used by many chemists for more than two centuries but over the last 30 years, the great significance of graph theory in chemistry has been extensively acknowledged. In a chemical graph, the vertices can symbolize atoms, orbitals, bonds, collection of atoms, molecules, or group of molecules and the edges symbolize the connection between chemical substances and are used to describe reactions, chemical bonds, kinetic models, reaction mechanisms, or other interactions and alterations of the chemical substances. Benzene is the archetypal aromatic molecule, showing exceptional chemical stability as compared with other unsaturated hydrocarbons. Fullerenes are a unique family of carbon-based cage molecules having remarkable properties. Fullerene has potential applications in medicinal chemistry due to its size, hydrophobicity, three-dimensionality, and electronic configurations. In this paper, we will calculate the fault-tolerant partition dimension of classes of fullerene graphs, n-linear benzene graph and partition dimension of polyphenyl chains. [ABSTRACT FROM AUTHOR]

The k -partition dimension of graphs was introduced as generalization of partition dimension of graphs by Estrado–Moreno in 2020. Its applications can be seen in several areas including robot navigation, network designing, image processing and chemistry. Rotationally symmetric graphs are important in computer network topologies due to uniform rate of data transformation to all nodes. In this paper, we compute the lower bound of the 2 -partition dimension of r -regular graphs for r ≥ 3 and also calculate the 2 -partition dimension of the families of rotationally symmetric graphs, namely, prism graph and n -sunlet graph. [ABSTRACT FROM AUTHOR]

In this paper, we compute first Zagreb index (coindex), second Zagreb index (coindex), third Zagreb index, first hyper-Zagreb index, atom-bond connectivity index, fourth atom-bond connectivity index, sum connectivity index, Randić connectivity index, augmented Zagreb index, Sanskruti index, geometric-arithmetic connectivity index and fifth geometric-arithmetic connectivity index of the line graphs of Banana tree graph and Firecracker graph.

In this paper, we study 3–total edge product cordial (3–TEPC) labeling which is a variant of edge product cordial labeling. We discuss Web, Helm, Ladder and Gear graphs in this context of 3–TEPC labeling. We also discuss 3–TEPC labeling of some particular examples with corona graph.

Carbon nanotubes have been actively studied for their applications in filtering of polycyclic aromatic hydrocarbons, sensors, electronics, stretchable devices, and energy storage. The large structures of nanotubes are studied using their induced subgraphs. The partition dimension focusses on the unique distance-based representation of nodes with respect to partition sets such that each node is exclusively identified. The partition and 2 − partition dimension of induced subgraphs of certain nanotubes have been computed in this article. [ABSTRACT FROM AUTHOR]

Let JG denote the binomial edge ideal of a connected undirected graph on n vertices. This is the ideal generated by the binomials xiyj − xjyi, 1 ≤ i < j≤ n, in the polynomial ring S = K[x1, . . . , xn, y1, . . . , yn] where {i, j} is an edge of G. We study the arithmetic properties of S/JG for G, the complete bipartite graph. In particular we compute dimensions, depths, Castelnuovo-Mumford regularities, Hilbert functions and multiplicities of them. As main results we give an explicit description of the modules of deficiencies, the duals of local cohomology modules, and prove the purity of the minimal free resolution of S/JG.

Let Λ = {B1, B2, . . ., Bl} be an ordered l-partition of a connected graph G(V (G), E(G)). The partition representation of vertex x with respect to Λ is the l-vector, r(x|Λ) = (d(x, B1), d(x, B2), . . ., d(x, Bl)), where d(x, B) = min{d(x, y)|y ∈ B} is the distance between x and B. If the l - vectors r(x|Λ), for all x ∈ V (G) are distinct then l - partition is called a resolving partition. The least value of l for which there is a resolving l - partition is known as the partition dimension of G symbolized as pd(G). In this paper, the partition dimension of circulant graphs Cn(1, 2, 3, 4) is computed for n ≥ 8 as, pd(Cn(1, 2, 3, 4)) = { if 8 ≤ n ≤ 9; { 6, if n = 10; {5, if n ≥ 11. [ABSTRACT FROM AUTHOR]