Further results on edge irregularity strength of graphs

A vertex k-labelling φ : V (G) −→ {1, 2, . . . , k} is called irregular k-labeling of the graph G if for every two different edges e and f , there is wφ(e) 6= wφ(f); where the weight of an edge is given by e = xy ∈ E(G) is wφ(xy) = φ(x) + φ(y). The minimum k for which the graph G has an edge irregular k-labelling is called edge irregularity strength of G, denoted by es(G). In the paper, we determine the exact value of the edge irregularity strength of caterpillars, n-star graphs, (n, t)-kite graphs, cycle chains and friendship graphs.


Introduction and preliminary results
The graph labeling has caught the attention of many authors and many new labeling results appear every year.This popularity is not only due to the mathematical challenges of graph labeling, but also for the wide range of its application, for instance X-ray, crystallography, coding theory, radar, astronomy, circuit design, network design and communication design.Bloom and Golomb studied applications of graph labelings to other branches of science [10,11].
All the graphs in this paper are finite, undirected and simple.For a graph G, the V (G) and E(G) denote the vertex set and edge set, respectively.A labeling of a graph G is any mapping that sends some set of graph elements to a set of non-negative integers.If the domain is vertex set or the edge set, the labeling is called vertex labelings or edge labelings, respectively.Moreover, if the domain is V (G) ∪ E(G), then the labeling is called a total labeling.Thus for an edge k-labeling, φ : E(G) −→ {1, 2, . . ., k}, the associated weight of a vertex x ∈ V (G) is where the sum is taken over all the vertices y adjacent to x. Chartrand et al. in [6] introduced edge k− labeling of a graph G such that w φ (x) = w φ (y) for all vertices x, y ∈ V (G) with x = y.Such labelings were called irregular assignments and the irregularity strength s(G) of a graph G is known as the minimum k for which G has irregular assignments using labels atmost k.Some results on irregularity strength s(G) of a graph G can be found in [1,3,6,7,8,12,13,14,15,16,17,18,19,20,21,22,23].Let φ be a vertex labeling of a graph G. Then we define the edge weight of xy ∈ E(G) to be w(xy) = φ(x) + φ(y).A vertex labeling φ : V (G) → {1, 2, . . ., k} is called k− labeling.Ali et al. in [2] introduced vertex k− labeling φ of a graph G such that w φ (e) = w φ (f ) for every two different edges e and f .Such a labeling were called an edge irregular k− labeling of the graph G.The minimum k for which the graph G has an edge irregular k− labeling is called the edge irregularity strength of G, denoted by es(G).They gives a lower bound of the parameter es(G) and determine the exact values of the edge irregularity strength for several family of graphs namely, paths, stars, double stars and cartesian product of two paths.

Theorem 1.1 ([2]
).Let G be simple graph with maximum degree ∆ = ∆(G).Then In this paper, we we determine the exact value of edge irregularity strength of we determine the exact value of the edge irregularity strength of caterpillars, n-star graphs, (n, t)-kite graphs, cycle chains and friendship graphs.

Main results
Let P n be a path on n vertices and let P n (k) be the graph which is obtained by attaching k edges to each vertex of P n .Then P n (k) is a caterpillar graph.The vertex set V (P n (k)) and edge set E(P n (k)) of this caterpillar graph Theorem 2.1.Let P n (k) be the caterpillar graph.If n is even, then es(P n (k)) = n(k+1) 2 .

2
. To prove the equality, it suffices to prove the existence of an edge irregular www.ijc.or.id Further results on edge irregularity strength of graphs | M. Imran, A. Aslam, S. Zafar and W. Nazeer , i ≡ 1 (mod 2).
Since w φ 1 (u i u i+1 ) = (k+1)i+1 and w φ 1 (u i u ij ) = k(i−1)+i+j for 1 ≤ i ≤ n and 1 ≤ j ≤ k, the weights of the edges under the labeling φ 1 successively attain values 2, 3, . . ., n(k +1).We can see that all vertex labels are at most n(k+1) 2 and edge weights are distinct for all pairs of distinct edges.Therefore the labeling φ 1 is suitable edge irregular n(k+1) The gluing together of identical cycles appears in various guises in the literature.But the construction of chains of cycles, with adjacent cycles sharing a single common vertex, is not prevalent.For this reason, we require the following definition.The graph C 2 n results from attaching two ncycles together at a single shared vertex.Continuing in this manner, we define C 3 n by attaching a third n-cycle to one of the n-cycles of C 2 n in a similar uniform manner so that the cycle containing two shared vertices consists of two identical n 2 -paths.Recursively, the graph C m n consists of a chain of m consecutive n-cycles.We refer to each of the graphs in this family as a cycle chain.
. It is not difficult to see that all vertex labels are at most mn 2 + 1 and the weights of the edges are pairwise distinct.Thus the vertex labeling φ 2 is an mn 2 + 1-labeling.Truszczynski [4] defines a dragon as a graph obtained by joining a cycle graph C n to a path P t of length t with a bridge.Kim and park [19] call them (n, t)− kites.Next theorem gives the exact value of the edge irregularity strength for (n, t)− kite.
Proof.Let G = (n, t)− kite graph, the vertex set of G is and the edge set of G is For the converse, we define a vertex n+t+1 2 -labeling φ 3 as follows: Case 1.If n = 2k and k ≡ 0 (mod 2), then we define φ 3 : V (G) → {1, 2, . . ., n+t+1 2 } as , the weights of the edges under the labeling φ 3 successively attain values 2, 3, . . ., n + t + 1.We can see that all vertex labels are at most n+t+1 2 and the edge weights are distinct for all pairs of distinct edges.Therefore the labeling φ 3 is a suitable edge irregular n+t+1 2 -labeling.Case 2. If n = 2k and k ≡ 1 (mod 2), then we define . It is not difficult to see that all vertex labels are at most n+t+1 We can see that all vertex labels are at most n+t+1 2 and the edge weights are distinct for all pairs of distinct edges.Therefore the labeling φ 3 is a suitable edge irregular n+t+1 2 -labeling.Hence, es(G) = n+t+1 2 .
In [7] Seoud and El Sakhawi introduced the following operation of graphs.The symmetric product , where P n is a path of order n and K * 2 is a null graph of order 2.
2 is a graph of order 2n and size 4n − 4.
For the converse, we define a suitable edge irregular labeling φ 4 : V (G) → {1, 2, . . ., 4n−3 2 } as follows: ), the weights of the edges under the labeling φ 4 successively attain values 3, 4, . . ., 4n − 2. We can see that all vertex labels are at most 4n−3 2 and the edge weights are distinct for all pairs of distinct edges.Therefore the labeling φ 4 is a suitable edge irregular Proof.The vertices of C t 4 are identified as follows: the common vertex of each cycle is identified as u.The remaining vertices of cycle C i are identified as c i,1 , c i,2 , c i,3 if we complete the cycle moving clockwise from the vertex u to itself.Now, for 1 ≤ i ≤ t we construct the function φ 5 : V (C t 4 ) → {1, 2, . . ., 2t + 1} as follows: One can see observe the labeling φ 5 is an edge irregular 2t+1-labeling, which implies the assertion.
Let T (n, k) be a graph obtained by connecting a vertex v to the central vertices of n copies of star on k vertices.In particular, n copies of star on k + 1 vertices shares a common single vertex v.We call T (n, k) a n− star graph.The vertex set V (T (n, k)) and edge set An exact value of the edge irregularity strength of n-star graph is given by the following theorem.Proof.We define a suitable edge irregular labeling φ 6 : V (T (n, k)) → {1, 2, . . ., nk+1 2 } as follows: The remaining vertices of T (n, k) are labeled depending on whether n ≡ 0 (mod 2) or n ≡ 1 (mod 2).Case 1.If n ≡ 0 (mod 2), then we define φ 6 as, We can see that all vertex labels are at most nk+1 2 = nk 2 + 1 and edge weights are distinct for all pairs of distinct edges.Therefore the labeling φ 6 is suitable edge irregular nk 2 + 1 labeling.
Hence es(T (n, k)) = nk 2 + 1 Case 2.1.If n ≡ 1 (mod 2) and n = k, then we define φ 6 as, We can see that all vertex labels are at most nk+1 2 = nk+1 2 and edge weights are distinct for all pairs of distinct edges.Therefore the labeling φ 6 is suitable edge irregular nk+1 2 labeling.Hence es(T (n, k)) = nk+1 2 .Case 2.2.If n ≡ 1 (mod 2) and n > k, then we define φ 6 as, We can see that the labeling φ 6 is an edge irregular nk+1 2 -labeling.Case 2.3.If n ≡ 1 (mod 2) and n < k, then we define φ 6 as, 2 and j < n+k We can see that all vertex labels are at most nk+1 2 and edge weights are distinct for all pairs of distinct edges.Therefore the labeling φ 6 is suitable edge irregular nk+1