A new generalization of edge-irregular evaluations

: Consider a simple graph G = ( V , E ) of size m with the vertex set V and the edge set E . A modular edge-irregular total k -labeling of a graph G is a labeling scheme for the vertices and edges with the labels 1 , 2 , . . . , k that allows the modular weights of any two di ff erent edges to be distinct, where the modular weight of an edge is the remainder of the division of the weight (i.e., the sum of the label of the edge itself and the labels of its two end vertices) by m . The maximal integer k , minimized over all modular edge-irregular total k -labelings of the graph G is called the modular total edge-irregularity strength. In the paper, we generalize the approach to edge-irregular evaluations, introduce the notion of the modular total edge-irregularity strength and obtain its boundary estimation. For certain families of graphs, we investigate the existence of modular edge-irregular total labelings and determine the precise values of the modular total edge-irregularity strength in order to prove the sharpness of the lower bound.


Introduction
Let G = (V, E) be a simple graph with vertex set V and edge set E. A labeling of a graph is a map that carries graph elements to the positive or non-negative integers.If the domain is the vertex set alone or the set of all vertices and edges, the labelings are respectively called vertex labelings or total labelings.Thus, for a vertex k-labeling ω : V(G) → {1, 2, . . ., k}, the associated weight of an edge e = xy ∈ E(G) is w ω (xy) = ω(x) + ω(y).A vertex labeling ω is called edge-irregular if all of the edges have distinct weights.The maximal integer k minimized over all edge-irregular k-labelings is known as the edge-irregularity strength of G, es(G).
The notion of the edge-irregularity strength was introduced by Ahmad et al. in [1], and there is an estimated lower bound of this graph invariant for all simple graphs with maximum degree ∆(G) in the following form:

2
, ∆(G) . (1.1) For certain families of graphs, specifically paths, stars, double stars and the Cartesian product of two paths [1], as well as for complete m-ary trees [2], Toeplitz graphs [4], plane graphs [22] and complete m-partite graphs [23], the exact value of the edge-irregularity strength has been determined.Koam et al., in [16], introduced a modular edge-irregular labeling as a modification of the modular irregular labeling defined in [8].
For a graph G = (V, E) of size m, a function ω : V(G) → {1, 2, . . ., k} is said to be a modular edge-irregular k-labeling if the edge-weight function µ : E(G) → Z m defined by µ(xy) = w ω (xy) = ω(x) + ω(y) is bijective and called the modular edge-weight of the edge xy, where Z m is the group of integers modulo m.The modular edge-irregularity strength, denoted as mes(G), is defined as the minimum k for which G has a modular edge-irregular labeling by using a number of labels of at most k.If there is no such labeling for the graph G, then the value of mes(G) is defined as ∞.
Let us note that Muthugurupackiam and Ramya in [18,19] defined the even (odd) modular edgeirregular labelings, where the set of modular edge-weights contains only even (odd) integers.
Definitely, every modular edge-irregular k-labeling of a graph is also its edge-irregular k-labeling.This shows a relationship between the edge-irregularity strength and the modular edge-irregularity strength.Thus, for any simple graph G, the following holds: es(G) ≤ mes(G). (1. 2) The relationship given by (1.2) gives a lower bound of the parameter mes(G).The existence of modular edge-irregular labelings for several families of graphs is investigated in [16], particularly for paths, stars, cycles, caterpillars, n-suns and friendship graphs, and the precise value of the modular edge-irregularity strength that proves the sharpness of the lower bound presented in (1.2) is determined.
For a graph G, the authors of [7] define a labeling ρ : V(G) ∪ E(G) → {1, 2, . . ., k} as an edgeirregular total k-labeling if, for every two different edges xy and x ′ y ′ of G there is wt ρ (xy) wt ρ (x ′ y ′ ), where the total edge-weight of an edge xy is defined as wt ρ (xy) = ρ(x) + ρ(xy) + ρ(y).The total edgeirregularity strength of a graph G, denoted as tes(G), is defined as the minimum k for which G has an edge-irregular total k-labeling.
One can see that, if ω : V(G) → {1, 2, . . ., k} is an edge-irregular k-labeling of a simple graph G with es(G) = k and we extend this labeling to the total labeling ρ of G such that for every e ∈ E(G), then the resulting labeling ρ is an edge-irregular total labeling of G. Thus, for any simple graph G, tes(G) ≤ es(G). (1.3) A lower bound on the total edge-irregularity strength of a simple graph with maximum degree ∆(G) is given in [7] as follows: (1.4) Ivančo and Jendrol ' [13] posed the conjecture that, for an arbitrary graph different from K 5 , the total edge-irregularity strength is exactly the lower bound presented by (1.4).
The conjecture of Ivančo and Jendrol ' has been verified for trees [13], for complete graphs and complete bipartite graphs [14], for the Cartesian product of two paths P n □P m [15], for the corona product of a path with certain graphs [20] and for large dense graphs with 2 in [10].Some other results on the total edge-irregularity strength can be found in [3,5,6,9].
An interesting modification of the irregularity strength of graphs is a strength of graphs.We refer the reader to [12] which deals with this graph invariant and presents a lower bound for the strength of a graph in terms of its independence number.
A total labeling ρ : is bijective and called the modular total edge-weight of the edge xy.The modular total edge-irregularity strength of a graph G, denoted as mtes(G), is known as the maximal integer k, minimized over all modular edge-irregular total k-labelings, and it is rated to ∞ if no such function is possible.
In this paper, we are dealing with a boundary estimation of the modular total edge-irregularity strength.For certain families of graphs, we investigate the existence of modular edge-irregular total k-labelings and determine the precise values of the modular total edge-irregularity strength in order to prove the tightness of the presented lower bound.

Main results
Since every modular edge-irregular total k-labeling of a graph G is also its edge-irregular total k-labeling then we have tes(G) ≤ mtes(G). (2.1) Extending a modular edge-irregular k-labeling to the modular edge-irregular total k-labeling (similar to the above construction for extending an edge-irregular k-labeling to the edge-irregular total k-labeling) gives the following relationship: In general, the converse of (2.1) does not hold.The next theorem gives a condition when an edgeirregular total k-labeling of a graph is also its modular edge-irregular total k-labeling.
Theorem 2.1.Let G be a graph with tes(G) = k.If the total edge-weights under a corresponding edge-irregular total k-labeling constitute a set of consecutive integers, then The exact values of the total edge-irregularity strength for cycles and stars are determined as follows.
Theorem 2.3.[7] Let K 1,n be a star on n + 1 vertices, n ≥ 2.Then, tes(K 1,n ) = n+1 2 .From the previous theorems, it follows that the lower bound of the total edge-irregularity strength in (1.4) is sharp.In addition, it is shown in [7] that, under the condition of the described edge-irregular total n+2 3 -labeling for cycles and the edge-irregular total n+1 2 -labeling for stars, the corresponding total edge-weights in both two considered cases constitute the set of consecutive integers.Then, with respect to Theorem 2.1, we have the following corollaries.Corollary 2.2.Let K 1,n be a star on n + 1 vertices, n ≥ 2.Then, mtes(K 1,n ) = n+1 2 .The previous corollaries imply the sharpness of the lower bound of the modular total edgeirregularity strength given in (2.1).
In [17], an n-sun S(n) on 2n vertices is defined as a cycle C n with an edge terminating in a vertex of degree 1 attached to each vertex.Koam et al.,in [16], proved that es(S(n)) = mes(S(n)) = n + 1. ( Now, we present the modular total edge-irregularity strength for n-suns.is an upper bound for the total edge-irregularity strength of n-sun, we describe a total 2n+2 3 labeling for S(n).
Let k = 2n+2 3 .For n ≥ 6, we construct the function ρ as follows: Evidently, the total labeling ρ is a k-labeling.Moreover, given the total labeling ρ for weights of the edges of S(n), we have It is a matter of performing a routine check to see that the determined total edge-weights constitute a sequence of consecutive integers from 3 up to 2n + 2. It means that the total labeling ρ is an edgeirregular total 2n+2 3 -labeling.Figure 1 shows the optimal edge-irregular total labelings for S(3), S(4) and S( 5), where the total edge-weights in all cases form a sequence of consecutive integers.Thus, with respect to Theorem 2.1, we get that tes(S(n)) = mtes(S(n)) = 2n+2 3 for any n ≥ 3.This concludes the proof.□ A friendship graph f n , n ≥ 1, can be visualized as n triangles sharing a common central vertex and is otherwise independent, where |V( f n )| = 2n + 1 and |E( f n )| = 3n.For the friendship graph f n of order 2n + 1, it has been proved in [16] that According to (2.2), these results give an upper bound of the modular total edge-irregularity strength for f n .The existence of edge-irregular total labelings of the friendship graph was investigated in [7], wherein the authors proved that tes( f n ) = 3n+2 3 .Since the described edge-irregular total 3n+2 3 labeling of the friendship graph f n , n ≥ 1, allocates the corresponding total edge-weights from the set of consecutive integers {3, 4, . . ., 3n + 2}, then, from Theorem 2.1, we get the next corollary.
Corollary 2.3.Let f n be a friendship graph of size 3n, n ≥ 1.Then, mtes( A wheel W n , n ≥ 3, is a graph obtained by joining all vertices of a cycle C n to a further vertex called the center.Haryeni et al. investigated the modular edge-irregularity strength of wheels and proved the following result [11].
Theorem 2.5.[11] Let W n be a wheel of order n + 1.If n is odd and n ≥ 5, then They also showed that the upper bound given in Theorem 2.5 is tight.From the relationship given by (2.2), we have an upper bound of the modular total edge-irregularity strength for W n for odd n.On the other hand, the authors of [7] present a construction of an edge-irregular total 2n+2 3 -labeling of the wheel with n + 1 vertices for all n ≥ 3, where the total edge-weights constitute the set of consecutive integers {3, 4, . . ., 2n + 2}.Given Theorem 2.1, we have the following corollary. .Partial results on the existence of edge-irregular total labelings for the circulant graph C n (1, 2), for n even, can be found in [21].
We will determine the exact value of the parameter mtes for each of the three classes of circulant graphs, namely, C n (1, 2), C n (1, 3) and C n 1, n 2 .Theorem 2.6.Let C n (1, 2) be a circulant graph on n vertices, n ≥ 5.Then, if n ≥ 6.

Corollary 2 . 4 . 3 .
Let W n be a wheel on n + 1 vertices, n ≥ 3.Then, mtes(W n ) = 2n+2In the last part of the paper, we focus on circulant graphs and study the existence of the modular edge-irregular total labelings for three classes of this family of graphs.Let n, m and a 1 , a 2 , . . ., a m be positive integers.For  1 ≤ a 1 < a 2 < • • • < a m ≤ n2 , the circulant graph C n (a 1 , a 2 , . . ., a m ) is a regular graph with the vertex set V = {v 0 , v 1 , . . ., v n−1 } and the edge set E = {v i v i+a j : i = 0, 1, . . ., n − 1, j = 1, 2, . . ., m}, where indices are taken modulo n.If a m < n 2 , then C n (a 1 , a 2 , . . ., a m ) is a 2m-regular circulant graph with mn edges.If a m = n 2 , then the circulant graph is a (2m − 1)-regular one of size n(2m−1) 2

3 .
where indices are taken modulo n.For n ≥ 5, the circulant graph C n (1, 2) is a 4-regular graph with n vertices and 2n edges.Put k = 2n+2 According to the lower bound given in (1.4), we have that tes(C n (1, 2)) ≥ k.