Weighted Graph Irregularity Indices Defined on the Vertex Set of a Graph

Performing comparative tests, some possibilities of constructing novel degree- and distance-based graph irregularity indices are investigated. Evaluating the discrimination ability of different irregularity indices, it is demonstrated (using examples) that in certain cases two newly constructed irregularity indices, namely IR DE A and IR DE B , are more selective.


Introduction
Only connected graphs without loops and parallel edges are considered in this study. For a graph G with n vertices and m edges, V(G) and E(G) denote the sets of vertices and edges, respectively. Let d(u) be the degree of vertex u of G. Let uv be an edge of G connecting the vertices u and v. Let Δ Δ(G) and δ δ(G) be the maximum and the minimum degrees, respectively, of G. In what follows, we use the standard terminology in graph theory; for notations not de ned here, we refer the readers to the books [1,2].
For a connected graph G, the set of numbers n j of vertices with degree j is denoted by n j n j (G): n j > 0, 1 ≤ j ≤ Δ}. For simplicity, the numbers n j (G) are called the vertex-parameters of graph G. For two vertices u, v ∈ V(G), the distance d (u, v) between u and v is the number of edges in a shortest path connecting them.
Two connected graphs G 1 and G 2 are said to be vertexdegree equivalent if they have an identical vertex-degree sequence. Certainly, if G 1 and G 2 are vertex-degree equivalent, then their vertex-parameters sets satisfy the equation n j (G 1 ) n j (G 2 ) for every j. A graph is called k-regular if all its vertices have the same degree k. A graph which is not regular is called a nonregular graph. A connected graph G is said to be bidegreed if its degree set consists of only two elements, where a degree set of G is the set of all distinct elements of its degree sequence.

Preliminary Considerations
A topological index TI of a graph G is any number associated with G (in some way) provided that the equation TI(G) TI(G ′ ) holds for every graph G ′ isomorphic to G. A lot of existing topological indices are degree-and distance-based ones [3][4][5]. Graph irregularity indices form a notable subclass of the class of traditional topological indices; where a topological index TI of a (connected) graph G is called a graph irregularity index if TI(G) ≥ 0, and TI(G) 0 if and only if graph G is a regular graph. Details about the existing graph irregularity indices can be found in [6,7]. e readers interested in the general concept of irregularity in graphs may consult the book [8].
In several situations, it is crucial to know how much irregular a given graph is; for example, see [9,10] where irregularity measures are used to predict physicochemical properties of chemical compounds, and see [11][12][13][14] for some applications of irregularity measures in network theory.
Most of the existing irregularity indices used in mathematical chemistry are degree-based irregularity indices.
ere exist irregularity indices which form a particular subset Φ of the set of degree-based irregularity indices; we say that an irregularity index ϕ belongs to the set Φ if for every pair of vertex-degree equivalent graphs G 1 and G 2 , the equation ϕ(G 1 ) � ϕ(G 2 ) holds. e most popular topological indices that are used in defining degree-based irregularity indices, are the first and second Zagreb indices (see for example [15]), denoted by M 1 and M 2 , respectively, and the so-called forgotten topological index [15], denoted by F. e first and second Zagreb indices of a graph G are defined as and the forgotten topological index is defined as ere exist numerous degree-based graph irregularity indices in literature, some of them are listed below. e variance Var is a degree-based graph irregularity index introduced by Bell [16]. e variance Var of a graph G of order n and size m is defined as We also consider the following four irregularity indices: It is remarked here that, except IR 2 , all the irregularity indices formulated above belong to the set Φ.

Weighted Irregularity Indices Defined on the Vertex Set of a Graph
In this section, we consider irregularity indices defined on the set of vertices of a graph G. e majority of these indices are weighted degree-and distance-based topological indices. Most of them may be considered as extended versions of the Wiener index; for example, see [17]. Let us consider the weighted vertex-based topological index of a graph G formulated as where Z(u, v) and W(u, v) are appropriately selected nonnegative 2-variable symmetric functions; both of them are defined on the vertex set V(G) of G. For simplicity, we call the function W(u, v) as the weight function of G. By taking in Equation (8), we get the following graph irregularity index where p is a positive real number. Depending on the choice of the parameter p and the weight function W(u, v), various types of irregularity indices can be deduced. For instance, the choices p � 1 and W(u, v) � 1 lead to the so-called total irregularity of a graph G defined by It was introduced by Abdo et al. in [18]. Also, assuming that p � 2 and W(u, v) � 1, we have the irregularity index Irrt 2 (G), introduced in Ref. [19]: At this point, the following known proposition [19] concerning Irrt 2 needs to be stated.

Proposition 1. For every graph G with n vertices and m edges, it holds that
In Equation (9), by taking , we obtain the following irregularity index: Note that IR D is a weighted degree-and distance-based irregularity index. Although IR D is a new irregularity index which is not known in the literature, but we prove in the next proposition that this irregularity index can be written in the linear combination of the following two topological indices and where D G (u) is identical to the transmission Tr(u) of the vertex u ∈ V(G) and Gut(G) is the so-called Gutman index; for example, see [20].
Proof. Note that For the graph G, it holds [21] that where ω(u) is any quantity associated with the vertex u of G. By taking ω(u) � d 2 u in (19) and using the obtained identity in (18), we get □ Remark 1. From Proposition 2, it follows that the inequality holds for every (connected) graph G, with equality if and only if G is regular.

Remark 2.
Because IR D is a weighted version of the irregularity index Irrt 2 , it is expected that its discrimination power is better than that of Irrt 2 . (20), one can establish another irregularity index IRQ defined by

Remark 3. Based on identity Equation
As Gut(G) > 1/2 for every (connected) graph of order at least 3, one has

Discriminating Ability of Novel Weighted Irregularity Indices
For comparing the discrimination ability of the irregularity indices IR D and IRQ with the traditional degree-based irregularity indices Var, IR 1 , IR 2 , and IR 3 , we use the 6vertex graphs G i (i � 1, 2, 3, 4) depicted in Figure 1. It is remarked here that the graphs shown in Figure 1 belong to the family of connected threshold graphs, and graph G 1 is isomorphic to the connected 6-vertex antiregular graph (for example, see [22,23]). For the four graphs depicted in Figure 1, computed values of preselected topological indices M 1 , M 2 , F, and corresponding irregularity indices are summarized in Tables 1 and 2.
Comparing irregularity indices listed in Tables 1 and 2, the following conclusions can be drawn. Among the four tested graphs, the index G 1 achieves the maximum value (that is, 249) of IR 3 . e irregularity indices IR 1 and IR 2 are maximum for the graph G 2 (namely, IR 1 (G 2 ) ≈ 0.375 and IR 2 (G 2 ) ≈ 0.5202). As it can be seen that Var(G 1 ) ≈ 1.667, while Var(G 2 ) � Var(G 3 ) � Var(G 3 ) ≈ 1.889 and that all the four graphs have the same value of Irrt 1 , which is 26. Also, the relation Irrt 2 (G) � n 2 Var(G) is confirmed for the considered graphs: and IR D(G 4 ) � 92, while the computed values of the irregularity index IRQ are different for all four graphs. From these observations, one can conclude that the degree variance Var, the total irregularity index Irr 1 , together with the irregularity indices Irr 2 , and IR D have a limited discrimination ability for the considered four graphs.

Novel Irregularity Indices Constructed by
Using the External Weight Concept e weight function W(u, v) included in (9) can be considered as an "internal" weight function. Introducing the external weight concept, one can construct novel irregularity indices. By using them, the original sequence of previously determined irregularity values can be appropriately modified for a given set of graphs considered.
By definition, an external weight EW(G) for a graph G is a positive-valued topological index computed as a function of one or more traditional topological indices. By means of an external weight EW(G) a novel irregularity index IRE(G) can be created as defined below: where IR(G) is an arbitrary irregularity index. By appropriately selected external weights EW(G), one can establish several different versions of irregularity indices IRE(G) satisfying some restrictions or desired expectations. As an Figure 1: Four 6-vertex nonregular graphs selected for tests.
Journal of Mathematics example, consider the three external weights defined for a graph G of order n and size m as follows: Using the three external weights listed above, the following irregularity indices of new type are obtained: (30) For graphs shown in Figure 1, the computed external weights and the corresponding irregularity indices are summarized in Table 3.
Comparing the computed irregularity indices mentioned in Table 3, one can conclude that the graph G 1 has the maximum irregularity indices IR DE A (G 1 ) � 10.8 and IR DE B (G 1 ) � 795.6, while the maximum value of the irregularity index IR DE C is attained by the graph G 2 where IR DE C (G 2 ) � 0.4211 (it should be emphasized here that the graph G 1 is identical to the 6-vertex connected antiregular graph, and it is usually desired that the connected antiregular graph attains the maximum value of an irregularity index among all connected graphs of a fixed order.) It is remarked here that the irregularity indices IRQ and IR DE C are identical to each other because

Additional Considerations
An interesting open problem can be formulated as follows: find a deterministic relationship between the following weighted bond-additive indices (see [24]).
and weighted atoms-pair-additive indices Depending on the definitions of the above irregularity indices, we observe that there exist graphs for which the mentioned relationship is perfect. As an example, when p � 1 and W(u, v) � d(u, v) then for the wheel graph W n of order n with n ≥ 5, one has where AL is the Albertson irregularity index [25]. e sigma index σ(G) of a graph G is defined (for example, see [26]) as is irregularity index is a natural generalization of the Albertson irregularity index. For the wheel graph W n of order n with n ≥ 5, the following identity holds: It is possible to construct a particular graph family for which the concept outlined above can be extended. For two graphs J 1 and J 2 with disjoint vertex sets, J 1 ∪ J 2 denotes the disjoint union of J 1 and J 2 . e join J 1 + J 2 of J 1 and J 2 is the graph obtained from J 1 ∪ J 2 by adding edges between every vertex of J 1 and every vertex of J 2 .
Proposition 3. Define the bidegreed graph H n of order n as follows:  Table 2: Computed topological indices of the four graphs shown in Figure 1.