Abstract
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 and , are more selective.
1. Introduction
Only connected graphs without loops and parallel edges are considered in this study. For a graph with vertices and edges, and denote the sets of vertices and edges, respectively. Let be the degree of vertex of . Let be an edge of connecting the vertices and . Let and be the maximum and the minimum degrees, respectively, of . In what follows, we use the standard terminology in graph theory; for notations not defined here, we refer the readers to the books [1, 2].
For a connected graph , the set of numbers of vertices with degree is denoted by . For simplicity, the numbers are called the vertex-parameters of graph . For two vertices , the distance between and is the number of edges in a shortest path connecting them.
Two connected graphs and are said to be vertex-degree equivalent if they have an identical vertex-degree sequence. Certainly, if and are vertex-degree equivalent, then their vertex-parameters sets satisfy the equation for every . A graph is called -regular if all its vertices have the same degree . A graph which is not regular is called a nonregular graph. A connected graph is said to be bidegreed if its degree set consists of only two elements, where a degree set of is the set of all distinct elements of its degree sequence.
2. Preliminary Considerations
A topological index of a graph is any number associated with (in some way) provided that the equation holds for every graph isomorphic to . A lot of existing topological indices are degree- and distance-based ones [3–5]. Graph irregularity indices form a notable subclass of the class of traditional topological indices; where a topological index of a (connected) graph is called a graph irregularity index if , and if and only if graph is a regular graph. Details about the existing graph irregularity indices can be found in [6, 7]. The 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 [11–14] for some applications of irregularity measures in network theory.
Most of the existing irregularity indices used in mathematical chemistry are degree-based irregularity indices. There exist irregularity indices which form a particular subset of the set of degree-based irregularity indices; we say that an irregularity index belongs to the set if for every pair of vertex-degree equivalent graphs and , the equation holds.
The 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 and , respectively, and the so-called forgotten topological index [15], denoted by . The first and second Zagreb indices of a graph are defined as
and the forgotten topological index is defined as
There exist numerous degree-based graph irregularity indices in literature, some of them are listed below.
The variance is a degree-based graph irregularity index introduced by Bell [16]. The variance of a graph of order and size is defined as
We also consider the following four irregularity indices:
It is remarked here that, except , all the irregularity indices formulated above belong to the set .
3. 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 . The 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 formulated aswhere and are appropriately selected non-negative 2-variable symmetric functions; both of them are defined on the vertex set of . For simplicity, we call the function as the weight function of . By takingin Equation (8), we get the following graph irregularity indexwhere is a positive real number. Depending on the choice of the parameter and the weight function , various types of irregularity indices can be deduced. For instance, the choices and lead to the so-called total irregularity of a graph defined by
It was introduced by Abdo et al. in [18]. Also, assuming that and , we have the irregularity index , introduced in Ref. [19]:
At this point, the following known proposition [19] concerning needs to be stated.
Proposition 1. For every graph with vertices and edges, it holds that
In Equation (9), by taking and , we obtain the following irregularity index:
Note that is a weighted degree- and distance-based irregularity index. Although 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 indicesandwhere is identical to the transmission of the vertex and is the so-called Gutman index; for example, see [20].
Proposition 2. For a (connected) graph , it holds that
Proof. Note thatFor the graph , it holds [21] thatwhere is any quantity associated with the vertex of . By taking in (19) and using the obtained identity in (18), we get
Remark 1. From Proposition 2, it follows that the inequalityholds for every (connected) graph , with equality if and only if is regular.
Remark 2. Because is a weighted version of the irregularity index , it is expected that its discrimination power is better than that of .
Remark 3. Based on identity Equation (20), one can establish another irregularity index defined byAs for every (connected) graph of order at least 3, one has
4. Discriminating Ability of Novel Weighted Irregularity Indices
For comparing the discrimination ability of the irregularity indices and with the traditional degree-based irregularity indices , , , and , we use the 6-vertex graphs 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 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 , , , 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 achieves the maximum value (that is, 249) of . The irregularity indices and are maximum for the graph (namely, and ). As it can be seen that , while and that all the four graphs have the same value of , which is 26. Also, the relation is confirmed for the considered graphs: and . Moreover, we have and , while the computed values of the irregularity index are different for all four graphs. From these observations, one can conclude that the degree variance , the total irregularity index , together with the irregularity indices , and have a limited discrimination ability for the considered four graphs.
5. Novel Irregularity Indices Constructed by Using the External Weight Concept
The weight function 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 for a graph is a positive-valued topological index computed as a function of one or more traditional topological indices. By means of an external weight a novel irregularity index can be created as defined below:where is an arbitrary irregularity index. By appropriately selected external weights , one can establish several different versions of irregularity indices satisfying some restrictions or desired expectations. As an example, consider the three external weights defined for a graph of order and size as follows:
Using the three external weights listed above, the following irregularity indices of new type are obtained: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 has the maximum irregularity indices and , while the maximum value of the irregularity index is attained by the graph where (it should be emphasized here that the graph 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 and are identical to each other because
6. 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 and then for the wheel graph of order with , one haswhere is the Albertson irregularity index [25].
The sigma index of a graph is defined (for example, see [26]) as
This irregularity index is a natural generalization of the Albertson irregularity index. For the wheel graph of order with , 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 and with disjoint vertex sets, denotes the disjoint union of and . The join of and is the graph obtained from by adding edges between every vertex of and every vertex of .
Proposition 3. Define the bidegreed graph of order as follows:where is an -regular graph and each is an -regular graph. It holds thatwhere is a modified version of the generalized Albertson irregularity index (see [27]).
Proof. We note thatObserve that for every pair of nonadjacent vertices , which implies thatand hence Equation (39) yields the desired result.
As an example concerning Proposition 3, consider the bidegreed graph of order 14 and size 59 constructed as follows:where is the (2-regular) cycle graph with 4 vertices, is the (3-regular) complete bipartite graph of order 6, and is the (3-regular) complete graph on 4 vertices (see Figure 2. The graph contains ten vertices of degree 7 and four vertices of degree 12. Note that if , then and , where , and both the vertices have the degree 7 in . Thus,and the desired conclusion holds.
Data Availability
The data about this study may be requested from the authors.
Conflicts of Interest
The authors declare that they have no conflicts of interest.