Further results on the radio number for some construction of the path, complete, and complete bipartite graphs.

Let G be a connected graph with d i a m ( G ) . Then d ( u , v ) indicates the distance of u and v in G. For any pair of distinct vertices u , v of G, mapping from c : V ( G ) → N such that d G ( u , v ) + | c ( u ) - c ( v ) | ≥ d i a m ( G ) + 1 . The maximum label assigned to any vertex of G under a radio labeling c is known as the span of c. The radio number r n ( G ) of G is defined as the minimum span among all possible radio labelings of G. This paper aims to determine the radio number r n ( c ) for specific constructed families of graphs with diameter 3, such as P 2 2 ( N ( m , n ) ) , K 3 , m ( K n c ) , K 3 , m ( n P 3 ) , K 3 , 3 ( 2 K 1 , n ) , and K 5 ( F 3 n ) .

A graph-theoretic approach to ring analysis: Dominant metric dimensions in zero-divisor graphs.

This article investigates the concept of dominant metric dimensions in zero divisor graphs (ZD-graphs) associated with rings. Consider a finite commutative ring with unity, denoted as R, where nonzero elements x and y are identified as zero divisors if their product results in zero (x.y=0). The set of zero divisors in ring R is referred to as L(R). To analyze various algebraic properties of R, a graph known as the zero-divisor graph is constructed using L(R). This manuscript establishes specific general bounds for the dominant metric dimension (Ddim) concerning the ZD-graph of R. To achieve this objective, we examine the zero divisor graphs for specific rings, such as the ring of Gaussian integers modulo m, denoted as Zm[i], the ring of integers modulo n, denoted as Zn, and some quotient polynomial rings. Our research unveils new insights into the structural similarities and differences among commutative rings sharing identical metric dimensions and dominant metric dimensions. Additionally, we present a general result outlining bounds for the dominant metric dimension expressed in terms of the maximum degree, girth, clique number, and diameter of the associated ZD-graphs. Through this exploration, we aim to provide a comprehensive framework for analyzing commutative rings and their associated zero divisor graphs, thereby advancing both theoretical knowledge and practical applications in diverse domains.

Computing dominant metric dimensions of certain connected networks.

In the studies of the connected networks, metric dimension being a distance-based parameter got much more attention of the researches due to its wide range of applications in different areas of chemistry and computer science. At present its various types such as local metric dimension, mixed metric dimension, solid metric dimension, and dominant metric dimension have been used to solve the problems related to drug discoveries, embedding of biological sequence data, classification of chemical compounds, linear optimization, robot navigation, differentiating the interconnected networks, detecting network motifs, image processing, source localization and sensor networking. Dominant resolving sets are better than resolving sets because they carry the property of domination. In this paper, we obtain the dominant metric dimension of wheel, gear and anti-web wheel network in the form of integral numbers. We observe that the aforesaid networks have bounded dominant metric dimension as the order of the network increases. In particular, we also discuss the importance of the obtained results for the robot navigation networking.

Computing the Laplacian spectrum and Wiener index of pentagonal-derivation cylinder/Möbius network.

The Laplacian spectrum significantly contributes the study of the structural features of non-regular networks. Actually, it emphasizes the interaction among the network eigenvalues and their structural properties. Let Pn(Pn') represent the pentagonal-derivation cylinder (Möbius) network. In this article, based on the decomposition techniques of the Laplacian characteristic polynomial, we initially determine that the Laplacian spectra of Pn contain the eigenvalues of matrices LR and LS. Furthermore, using the relationship among the coefficients and roots of these two matrices, explicit calculations of the Kirchhoff index and spanning trees of Pn are determined. The relationship between the Wiener and Kirchhoff indices of Pn is also established.

Symmetries and symmetry-breaking in arithmetic graphs.

In this paper, we study symmetries and symmetry-breaking of the arithmetic graph of a composite number m, denoted by Am. We first study some properties such as the distance between vertices, the degree of a vertex and the number of twin classes in the arithmetic graphs. We describe symmetries of Am and prove that the automorphism group of Am is isomorphic to the symmetric group Sn of n elements, for m=∏i=1npi. For symmetry-breaking, we study the concept of the fixing number of the arithmetic graphs and give exact formulae of the fixing number for the arithmetic graphs Am for m=∏i=1npiri under different conditions on ri.

Computational analysis for eccentric neighborhood Zagreb indices and their significance.

In this paper, a novel eccentric neighborhood degree-based topological indices, termed eccentric neighborhood Zagreb indices, have been conceptualized and its discriminating power investigated with regard to the predictability of the boiling point of the chemical substances. The discriminating power of the eccentric neighborhood Zagreb indices was compared with that of Wiener and eccentric connectivity indices. Some explicit results for those new indices for some graphs and graph operations such as join, disjunction, composition, and symmetric difference.