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 ) .

This work proves the local vertex anti-magic coloring of even regular circulant bipartite graphs C ( m ; L ) . Let G be either K r , r or K r , r - F , F is a 1-factor. Also, we discover the local vertex anti-magic coloring for union of bipartite graphs; join graphs G ∨ H , where H ∈ { O r , K r , C r , K r , s } ; and the upper bound of corona product G ⊙ O r . It was a problem Lau and Shiu (2023) [1] that: For any G 1 and G 2 , determine χ ℓ v a ( G 1 × G 2 ) . We give partial answer to this problem by proving the followings:1. χ ℓ v a ( C 2 m × C 2 n ) ;2. χ ℓ v a ( C 2 m + 1 × C 2 n + 2 ) ; and3. χ ℓ v a ( P 3 × H ) , where H ∈ { K r , K m , m } .

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.

A prime labeling of a simple undirected graph G is to assign unique integer labels from the set {1,2,...,|V(G)|} to each vertex such that any two adjacent vertices in the graph have labels that are relatively prime. Studying prime labeling in graphs can help us understand the structure and properties of graphs and prime labeling has potential applications in cryptography and network security. In this paper we investigate when some graphs that are constructed from wheels are prime graphs.

In this article, we find an α-valuation for disjoint union of some bipartite graphs and the tensor product of paths and even cycles.

Let G=(V,E) be a graph without isolated vertices and let |V(G)|=n and |E(G)|=m. A bijection π:V(G)∪E(G)→{1,2,....,n+m} is said to be local total anti-magic labeling of a graph G if it satisfies the conditions: (i.) for any edge uv, ω(u)≠ω(v), where u and v in V(G) (ii.) for any two adjacent edges e and e\', ω(e)≠ω(e\') (iii.) for any edge uv∈E(G) is incident to the vertex v, ω(v)≠ω(uv), where weight of vertex u is, ω(u)=∑e∈S(u)π(e), S(u) is the set of edges with every edge of S(u) one end vertex is u and an edge weight is ω(e=uv)=π(u)+π(v). In this paper, we have introduced a local total anti-magic labeling (LTAL) and the local total anti-magic chromatic number (LTACN). Also, we obtain the LTACN for the graphs Pn, K1,n, Fn and Sn,n.