On edge irregularity strength of cycle-star graphs

For a simple graph $G$, a vertex labeling $\phi:V(G) \rightarrow \{1, 2,\ldots,k\}$ is called $k$-labeling. The weight of an edge $uv$ in $G$, written $w_{\phi}(uv)$, is the sum of the labels of end vertices $u$ and $v$, i.e., $w_{\phi}(uv)=\phi(u)+\phi(v)$. A vertex $k$-labeling is defined to be an edge irregular $k$-labeling of the graph $G$ if for every two distinct edges $u$ and $v$, $w_{\phi}(u) \neq w_{\phi}(v)$. The minimum $k$ for which the graph $G$ has an edge irregular $k$-labeling is called the edge irregularity strength of $G$, written $es(G)$. In this paper, we study the edge irregular $k$-labeling for cycle-star graph $CS_{k,n-k}$ and determine the exact value for cycle-star graph for $3 \leq k \leq 7$ and $n-k \geq 1$. Finally, we make a conjecture for the edge irregularity strength of $CS_{k,n-k}$ for $k \geq 8$ and $n-k \geq 1$.


Introduction
Let G be a connected, simple, and undirected graph with vertex set V (G) and edge set E(G).By a labeling we mean any mapping that maps a set of graph elements to a set of numbers (usually positive integers), called labels.If the domain is the vertex-set (the edge-set), then the labeling is called vertex labelings (edge labelings).If the domain is V (G) ∪ E(G), then the labeling is called total labeling.Thus, for an edge k-labeling δ : E(G) → {1, 2, . . ., k} the associated weight of a vertex x ∈ V (G) is w δ (x) = δ(xy), where the sum is over all vertices y adjacent to x.
Chartrand et al. [9] defined irregular labeling for a graph G as an assignment of labels from the set of natural numbers to the edges of G such that the sum of the labels assigned to the edges of each vertex are different.The minimum value of the largest label of an edge over all existing irregular labelings is known as the irregularity strength of G, denoted by s(G).Finding the irregularity strength of a graph seems to be hard even for simple graphs, see e.g., [9,10].
Motivated by this, Baca et al. [7] investigated the irregularity strength of graphs, namely total edge irregularity strength, denoted by tes(G); and total vertex irregularity strength, denoted by tvs(G).Some results on the total edge irregularity strength and the total vertex irregularity strength can be found in [1,2,4,6,8].
Motivated by the work of Chartrand et al. [9], Ahmad et al. [3] introduced the concept of edge irregular k-labelings of graphs.
A vertex labeling ϕ : V (G) → {1, 2, . . ., k} is called k-labeling.The weight of an edge uv in G, written w ϕ (uv), is the sum of the labels of end vertices u and v, i.e., w ϕ (uv) = ϕ(u) + ϕ(v).A vertex k-labeling of a graph G is defined to be an edge irregular k-labeling of the graph G if for every two different edges u and v, w ϕ (u) ̸ = w ϕ (v).The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, written es(G).Over the last years, es(G) has been investigated for different families of graphs including trees with the help of algorithmic solutions, see [12,13,14,16,17,18,19,20,21,22,23].The most complete recent survey of graph labelings is [11].
Sedlar [15] introduced the concept of cycle-star graph as follows.The cycle-star graph, written CS k,n−k , is a graph with n vertices consisting of the cycle graph of length k and n − k leafs appended to the same vertex of the cycle.Let the vertex set and edge set of a cycle-star graph be

Main results
The following theorem in [3] determines the lower bound for the edge irregularity strength of a graph G. Theorem 3.1 Let G = (V, E) be a simple graph with maximum degree ∆(G).Then For two vertices u and v in a graph G, the distance d(u, v) from u to v is the length of a shortest u − v path in G.For a vertex v in a connected graph G, the eccentricity e(v) of v is the distance between v and a vertex farthest from v in G.The minimum eccentricity among the vertices of G is its radius and the maximum eccentricity is its diameter, which are denoted by rad(G) and diam(G), respectively.A vertex v in G is a central vertex if e(v) = rad(G).
We will make use of Theorem 3.1 to prove our main results.In the next theorem, we determine the exact value of the edge irregularity strength of cycle-star graph CS k,n−k for k = 3 and n − k ≥ 1.
Proof.Let G = CS k,n−k be a cycle-star graph, where k = 3 and n − k ≥ 1.Let us consider the vertex set and edge set of G.
Let v be the central vertex; v 1 , v 2 , . . ., v n−3 be leafs that are adjacent to v; and let v n−1 , v n−2 be the other vertices on the cycle C 3 .We have to show that es(G) = n − 1.From Theorem 3.1, we get the lower bound es(G) ≥ n − 1.
To prove the equality, it suffices to prove the existence of an edge irregular (n−1)− labeling.
Define a labeling on vertex set of G as follows: The edge weights are as follows: On the basis of above calculations we see that the edge weights are distinct for all pairs of distinct edges.Thus the vertex labeling ϕ is an edge regular (n − 1)− labeling.Therefore, es(G) = n − 1.
Let v be the central vertex; v 1 , v 2 , . . ., v n−4 be leafs that are adjacent to v; and let v n−1 , v n−2 , v n−3 be the other vertices on the cycle C 4 .We have to show that es(G) = n − 2. From Theorem 3.1, we get the lower bound es(G) ≥ n − 2.
To prove the equality, it suffices to prove the existence of an edge irregular (n−2)− labeling.
Define a labeling on vertex set of G as follows: The edge weights are as follows: On the basis of above calculations we see that the edge weights are distinct for all pairs of distinct edges.Thus the vertex labeling ϕ is an edge regular (n − 2)− labeling.Therefore, es(G) = n − 2.
We consider the following two cases.For the cycle-star graph CS 5,1 , let v be the central vertex; v 1 be the leaf adjacent to v; and let v 2 , v 3 , v 4 , v 5 be the other vertices on cycle C 5 .
The edge weights are as follows: On the basis of above calculations we see that the edge weights are distinct for all pairs of distinct edges.Thus the vertex labeling ϕ is an edge regular (n − 2)− labeling.Therefore, es(CS 5,1 ) = n − 2 for n = 6.
The edge weights are as follows: On the basis of above calculations we see that the edge weights are distinct for all pairs of distinct edges.Thus the vertex labeling ϕ is an edge regular (n − 3)− labeling.Therefore, es(G) = n − 3 for k = 5 and n ≥ 7.
We consider the following two cases.To prove the equality, it suffices to prove the existence of an edge irregular (n−3)− labeling.
For the cycle-star graph G = CS 6,n−6 , where n = 7, 8, let v be the central vertex; v 2 be the leaf adjacent to v; and let v 1 , v 3 , v 4 , v 5 , . . ., v n−1 be the other vertices on the cycle C 6 .
Let ϕ : V (G) → {1, 2, . . ., n − 3} be the vertex labeling such that ϕ The edge weights are as follows: On the basis of above calculations we see that the edge weights are distinct for all pairs of distinct edges.Thus the vertex labeling ϕ is an edge regular (n − 3)− labeling.Therefore, es(G) = n − 3 for n = 7, 8.
The edge weights are as follows: On the basis of above calculations we see that the edge weights are distinct for all pairs of distinct edges.Thus the vertex labeling ϕ is an edge regular (n − 4)− labeling.Therefore, es(G) = n − 4 for k = 6 and n ≥ 9.
We consider the following three cases.To prove the equality, it suffices to prove the existence of an edge irregular (n−3)− labeling.
For the cycle-star graph g = CS 7,1 , let v be the central vertex; v 1 be the leaf adjacent to v; and let v 2 , v 3 , v 4 , v 5 , v 6 , v 7 be the other vertices on the cycle C 7 .
The edge weights are as follows: The edge weights are as follows: On the basis of above calculations we see that the edge weights are distinct for all pairs of distinct edges.Thus the vertex labeling ϕ is an edge regular (n − 4)− labeling.Therefore, es(G) = n − 4 for n = 9, 10.We close with the following conjecture.

Conclusion
In this paper, we investigated the edge irregularity strength, as a modification of the well-known irregularity strength, total edge irregularity strength and total vertex irregularity strength.We obtained the exact values for edge irregularity strength of cycle-star graph CS k,n−k for 3 ≤ k ≤ 7 and n − k ≥ 1.Also, we conjectured the edge irregularity strength of cycle-star graph CS k,n−k for k ≥ 8 and n − k ≥ 1.The exact values of edge irregularity strength can be determined for graph operations, graph products, and graph powers also.