The Irregularity and Modular Irregularity Strength of Fan Graphs

For a simple graph G with no isolated edges and at most, one isolated vertex, a labeling φ:E(G)→{1,2,…,k} of positive integers to the edges of G is called irregular if the weights of the vertices, defined as wtφ(v)=∑u∈N(v)φ(uv), are all different. The irregularity strength of a graph G is known as the maximal integer k, minimized over all irregular labelings, and is set to ∞ if no such labeling exists. In this paper, we determine the exact value of the irregularity strength and the modular irregularity strength of fan graphs.


Introduction
It is well-known that a simple graph of an order of at least two must contain a pair of vertices with the same degree. However, a multigraph can be irregular, that is, each vertex can have a different degree. By Frieze et al. in [1], a natural question would be: What is the least number of edges we would need to add to a graph in order to convert a simple graph into an irregular multigraph?
Motivated by this question, Chartrand et al. in [2] introduced an edge k-labeling ϕ : E(G) → {1, 2, . . . , k} of a graph G (i.e., a mapping that assigns numbers 1, 2, . . . , k to the edges of G) with the property that the weights of the vertices are all different. The weight of a vertex v ∈ V(G) is defined as wt ϕ (v) = ∑ u∈N(v) ϕ (uv), where N(v) denotes the set of neighbors of v in G. Such labelings were called irregular assignments. Note that as the induced vertex weights are all distinct and the only edges of a graph are labeled with the numbers 1, 2, . . . , k this assignment can also be called vertex irregular edge k-labeling. The irregularity strength s(G) of a graph G is known as the maximal integer k, minimized over all irregular assignments. This means that the irregularity strength of a graph G is the minimum k for which a graph admits an irregular assignment using the number k as the largest edge label. If no such labeling of G exists, then s(G) = ∞. Clearly, the irregularity strength is finite only for graphs that contain, at most, one isolated vertex and no isolated edges. To view the irregularity strength via the degree-based problem, this graph invariant is connected to the maximal number of edges joining any pair of vertices in an irregular multigraph corresponding to the given graph G.
The lower bound of the irregularity strength is given in [2] in the form s(G) ≥ max n i +i−1 where n i denotes the number of vertices of degree i and ∆ is the maximum degree of the graph G. For d-regular graphs of order n, the lower bound (1) reduces to Faudree and Lehel in [3] showed that if G is a d-regular graph of order n, d ≥ 2, then s(G) ≤ n 2 + 9, and they conjectured that there exists an absolute constant C such that s(G) ≤ n d + C. For general graphs with no component of order at most 2, it is known that s(G) ≤ |V(G)| − 1, see [4,5]. This upper bound was gradually improved by Cuckler and Lazebnik in [6], Przybyło in [7], Kalkowski, Karonski, and Pfender in [8], and recently by Majerski and Przybyło in [9]. Other interesting results on the irregularity strength can be found in [1,10].
The exact value of the irregularity strength of particular families of graphs are known, where among them are paths, complete graphs [2], cycles, most of the complete bipartite graphs, Turan graphs [11], generalized Petersen graphs [12], circulant graphs [13], and trees [14]. For more results, see [15].
A natural modification of an irregular assignment is a modular irregular assignment introduced in [16]. Edge k-labeling ϕ : E(G) → {1, 2, . . . , k} of positive integers to the edges of a graph G of order n is called a modular irregular assignment of G if the weight function ϑ : V(G) → Z n defined by is bijective and is called as the modular weight of the vertex v, where Z n is the group of integers modulo n. The modular irregularity strength, ms(G), is defined as the minimum k for which G has a modular irregular assignment. If there is no such labeling for the graph G, then the value of ms(G) is defined as ∞.
In [16], a lower bound of the modular irregularity strength is established, and the exact values of this parameter for certain families of graphs, namely paths, cycles, stars, triangular graphs, and gear graphs are determined.
A fan graph F n , n ≥ 2 is a graph obtained by joining all vertices of path P n on n vertices to a further vertex, called the centre. Thus, F n contains n + 1 vertices, say, u 1 , u 2 , . . . , u n , w, and 2n − 1 edges, say, u i u i+1 , 1 ≤ i ≤ n − 1, and u i w, 1 ≤ i ≤ n.
In this paper, we determine the exact value of the irregularity strength and the modular irregularity strength of fan graphs F n of order n + 1. The rest of the article is organized as follows. First we deal with the irregularity strength of fan graphs. We describe a desired labeling scheme that proves the exact value of the irregularity strength of fan graphs. We describe a labeling scheme with symmetrical distribution of even weights and odd weights of vertices u i . We use this symmetrical distribution of the weights to prove that the weight of the centre w is always greater than the weights of u i . It proves that the labeling scheme is a desired vertex irregular edge labeling that proves the exact value of the irregularity strength of fan graphs. Next, by modifications of this irregular assignment we obtain labelings that imply the results for the modular irregularity strength of fan graphs.

Fan Graphs-The Irregularity Strength
The main result of this subsection is the following theorem. Theorem 1. Let F n , n ≥ 3, be a fan graph on n + 1 vertices. Then, , if n ≥ 3.
To prove the above-mentioned result, we present several lemmas. The first lemma gives a lower bound for the irregularity strength for the fan graphs. Lemma 1. Let F n , n ≥ 3, be a fan graph on n + 1 vertices. Then, Proof. According to the general lower bound (1), we have that the irregularity strength for the fan graphs s(F n ) ≥ n 3 for n ≥ 4. However, we can improve this bound. If we consider only vertices u i ∈ V(F n ), i = 1, 2, . . . , n, and assume that an edge labeling ϕ is the irregular assignment of F n with s(F n ) = k, then the smallest weight of each considered vertex is at least 2, and the largest weight admits the value at least n + 1, and at most 3k.
For n ≥ 3, we define the edge labeling ϕ in the following way: Now we prove that the above-defined labeling ϕ is an n+1 3 -labeling, and that the vertex weights induced by the labeling ϕ are all distinct.
The following lemma shows that under the edge labeling ϕ, the edge labels of F n are bounded from above. If n is odd, then If n is even, then It is easy to see that if n is odd, then the parameter b = n−1 6 + n+1 Proof. One can check that If n is even, then If n is odd, then Combining the previous, the result follows.

Lemma 4.
The centre w of the fan graph F n , under the labeling ϕ, admits the weight Proof. If we consider a triple of edges (u 3s+1 w, u 3s+2 w, u 3s+3 w) for s = 0, 1, 2, . . . , p, where p = n 6 when n ≡ 4 (mod 6) and p = n 6 − 1 otherwise, then the sum of the labels of edges for each triple is If we consider a triple of edges (u n−3r−1 w, u n−3r−2 w, u n−3r−3 w) for r = 0, 1, 2, . . . , q, where q = n 6 − 2 when n ≡ 0, 2 (mod 6) and q = n 6 − 1 otherwise, then the sum of labels of edges for each such triple is Next we consider six cases according to the residue of n modulo 6. Case 1. n ≡ 0 (mod 6).
Proof of Theorem 1. The fan graph F 2 is isomorphic to a cycle C 3 . It admits an irregular assignment with edge labels 1, 2, 3 and with the induced vertex weights 3, 4, 5. Thus, s(F 2 ) = s(C 3 ) = 3. According to Lemma 1, we have that s(F n ) ≥ n+1 3 for n ≥ 3. To prove the equality, it suffices to prove the existence of a vertex irregular edge n+1 3 -labeling of F n . For n ≥ 3, consider the edge labeling ϕ of F n defined by (4). From Lemma 2, it follows that ϕ is an n+1 3 -labeling. Lemma 3 proves that weights of the vertices u i , i = 1, 2, . . . , n, under the labeling ϕ successively attain values 2, 3, . . . , n + 1. Moreover, with respect to Lemma 4, we get that wt ϕ (w) > n + 1 for every n ≥ 3. Thus, the vertex weights are distinct for all pairs of distinct vertices. Therefore, the labeling ϕ is a suitable vertex irregular edge n+1 3 -labeling of F n . This concludes the proof.

The Modular Irregularity Strength of the Fan Graphs
Let us recall the following two lemmas.

Lemma 5. [16]
Let G be a graph with no component of order ≤ 2. Every modular irregular labeling of G is also its irregular assignment.
In general, the converse of the previous lemma does not hold. For example, the edge labeling of star K 1,3 with edge labels 1, 2, 3 is an irregular assignment with vertex weights 1, 2, 3, 6. However, this irregular labeling is not modular. If we label the edges of the star K 1,3 by labels 1, 2, 4, then we get a modular irregular assignment with modular vertex weights 0, 1, 2, 3, and ms(K 1,3 ) = 4.
The next statement gives a condition when an irregular assignment of a graph is also its modular irregular labeling. The following theorem gives a lower bound of the modular irregularity strength.

Theorem 2. [16]
Let G be a graph with no component of order ≤ 2. Then, s(G) ≤ ms(G). Now, we give the precise value of the modular irregularity strength for fan graphs F n , for n ≥ 2 even. Theorem 3. Let F n be a fan graph on n + 1 vertices with n ≥ 2 even. Then,

otherwise.
Proof. Let n = 2. We have already mentioned that the fan graph F 2 admits an irregular assignment with edge labels 1, 2, 3 and with vertex weights 3, 4, 5. From Lemma 6, it follows that the irregular assignment of F 2 is modular and ms(F 2 ) = 3. Let n = 8. Suppose that there exists a modular irregular 3-labeling ξ of F 8 . As the vertices u i , i = 1, 2, . . . , 8 are either of degree 2 or 3, the weights of these vertices under any 3-labeling is at least 2 (this can be realizable only on a vertex of degree 2 as the sum of edge labels 1 + 1) and is at most 9 (this can be realizable only on a vertex of degree 3 as the sum of edge labels 3 + 3 + 3). As all the vertices must have distinct modular weights, we get that the weights of the vertices u i , i = 1, 2, . . . , 8, constitute the sequence of consecutive integers from 2 up to 9. Thus, the modular weight 1 can be obtained only by the centre w. Moreover, it is easy to see that the weight of the centre cannot be 10 (must be at least 11 but at most 21). Thus, wt ξ (w) = 19 ≡ 1 (mod 9). Then, wt ξ (u i ) = 44, then Equation (5) gives which is a contradiction. Thus, there is no modular irregular 3-labeling for F 8 . Figure 1 shows an example of a modular irregular 4-labeling of the fan graph F 8 , where the modular weights are depicted in italic font. It is sufficient to consider the edge irregular n+1 3 -labeling ϕ defined by (4). By Lemma 3, under the labeling ϕ, the weights of all vertices u i ∈ V(F n ), i = 1, 2, . . . , n, successively assume values 2, 3, . . . , n, n + 1, and by Lemma 4, the weight of the centre vertex is wt ϕ (w) = (n+1)(n+2) 6 + 1. Since n ≡ 4 (mod 6), then n+2 6 is an integer and (n+1)(n+2) 6 + 1 ≡ 1 (mod n + 1). This implies that the labeling ϕ is a suitable modular irregular n+1
Observe that under the labeling ϕ, by Lemma 4, the centre of F n admits the weight wt ϕ (w) = n(n+3) 6 + 2. Since n ≡ 0 (mod 6), it follows that wt ϕ (w) ≡ 1 (mod n + 1) and the labeling ϕ is not modular irregular. Therefore, we need to modify the labeling ϕ. Figure 2 illustrates a modular irregular 3-labeling of the fan graph F 6 , where the modular weights are depicted in italic font. Proof. It is a matter of routine checking to see that under the edge labeling ϕ defined by (4), for n odd, the weights of the centre of F n listed in Lemma 4 are not congruent to 1 (mod n + 1). In order to preserve the property of the edge labeling ϕ that the weights of all vertices u i ∈ V(F n ), i = 1, 2, . . . , n, form the set {2, 3, . . . , n, n + 1}, and to attain the weights of the centre congruent to one (mod n + 1), we will construct some appropriate modifications of the labeling ϕ in a similar way as in the proof of Theorem 3.
Since we consider only n odd and moreover, n ≡ 1 (mod 4), then we distinguish the following three cases according to the residue of n modulo 12.
For n ≥ 15, we define an edge labeling ρ of F n such that: By a direct verification, we can detect that under the labeling ρ, all edge labels are at most n+1 3 , the weights of the vertices u i , 1 ≤ i ≤ n constitute a sequence of consecutive integers from 2 up to n + 1, and the weight of the centre determined by Lemma 4 is increased by n−3 6 . Consequently, wt ρ (w) = wt ϕ (w) + n−3 6 = n(n+3) 6 + 2 + n−3 6 = (n+3)(n+1) 6 + 1.
As n+3 6 is an integer, then wt ρ (w) ≡ 1 (mod n + 1). Case 2. n ≡ 7 (mod 12). Figure 3 depicts a modular irregular 3-labeling of the fan graph F 7 . The modular weights are again illustrated using italic font. For n ≥ 19, we define an edge labeling λ of F n in the following way: λ(u i u i+1 ) = ϕ(u i u i+1 ), for 1 ≤ i ≤ n−1 2 , i = 3, 3, for i = 3, By modifying an irregular assignment of the fan graph, we obtained modular irregular assignments and proved that , otherwise.
According to the given result, we get that the fan graphs are an example of graphs for which the irregularity strength and the modular irregularity strength are almost the same, up to a small case and one case excluded by the necessary condition for the modular irregularity strength to be finite. Thus, naturally, we conclude our paper with the following open problem.