IRREGULARITY OF MOLECULAR GRAPHS

A graph whose all vertices have equal degrees is said to be regular. If this is not the case, then the graph is irregular. Various measure of irregularity have been proposed. These are described and compared, with particular emphasis on molecular graphs.


Introduction
Let G = (V, E) be a (molecular) graph with vertex set V = V (G) and edge set E = E(G), having n = |V (G)| vertices and m = |E(G)| edges [26,32].Let V (G) = {v 1 , v 2 , . . ., v n }.For v i ∈ V (G), the degree of the vertex v i , denoted by d is the number of vertices adjacent to v. The degree sequence of the graph G is the non-increasing sequence of its degrees, such that At this point we recall that The number of vertices of the graph G whose degree is equal to i will be denoted by n i = n i (G).Then, of course, and according to Eq. ( 2), A graph G is is said to be regular if all its vertices have the same degree.Regular graphs played an outstanding role in the history of graph theory [11] and are still in the focus of interest of mathematicians.This is seen from the fact that numerous regular graphs have special names, such as the Petersen graph, Moore graph, Hoffman-Singleton graph, Shrikhande graph, Klein graph, Hall-Janko graph, Schläfli graph, etc. Important and much studied is the class of "strongly regular graphs".In mathematical chemistry, regular graphs occur has much less frequently.
Only after the the discovery of fullerenes and nanotubes, such graphs started to attract the attention of mathematical chemists.Irrespective of this, the fact is that the vast majority of molecular graphs are non-regular.
A graph which is not regular is said to be irregular.
It seems that Pál Erdős1 was the first who recognized that it is reasonable to ask about how irregular a non-regular graph is.In 1988, Erdős et al. published a paper [12] in which they asked "Which class of graphs is opposite to the regular graphs?" Such "opposite" graphs should be highly irregular.During the Second Krakow Conference on Graph Theory (1994), Erdős officially posed as an open problem the determination of the extreme size of highly irregular graphs of given order [40].
Evidently, a graph would be maximally irregular if all its degrees would differ, i.e., if Graphs with this property were named "perfect".However, Behzad and Chartrand [9] established that perfect graphs do not exist.What does exist are the "quasi-perfect" graphs (sometimes referred to as "antiregular graphs" [43]).These are graphs in which all vertices, except two, have different degrees.The respective result is [9]: For each n > 1, there is a unique connected quasi-perfect graph of order n.Its degree sequence is Note that the complement of a quasi-perfect graph is also quasi-perfect.However, the complement of a connected quasi-perfect graph is disconnected, and is therefore of no relevance for the present considerations.
Of these quasi-perfect graphs only those depicted in Fig. 1 are molecular graphs.
Fig. 1.The first four connected quasi-perfect graphs.With the exception of Q 1 , these may be considered as the most irregular molecular graphs.

Irregularity index
A simple and straightforward way of expressing the irregularity of a graph is via its irregularity index, Irr d , equal to the the number of distinct elements in the degree sequence, or in a more formal notation: This concept was, in an implicit manner, used in the early works [5,6,39,40], but was explicitly considered only quite recently [36,37,41,42].
In the case of molecular graphs, the irregularity index should be applied with due caution, or -better -not applied at all.Namely, by counting only the number of different values in the degree sequence d(G), Eq. ( 1 ).Yet, in G 3 all but two edges connect a degree two and a degree three vertex.Contrary to this, G 4 may be viewed as consisting of a regular graph of degree three joined by an edge to a regular graph of degree two.Intuitively, one would expect that G 3 is much more irregular than G 4 .
3 Structure-dependent irregularity indices measure of graph irregularity, say Irr(G), must satisfy the following requirements: (α) Irr(G) = 0 if and only if the (connected) graph G is regular.
The irregularity index Irr d described above, satisfies these requirements.However, bearing in mind the difficulties explained in the preceding section, another requirement for Irr needs to be added: (γ) The quantity Irr has to be defined so that its numerical value follows our intuitive feeling for "deviating from regularity".
It seems that the first such measure of graph irregularity is found in the seminal work of Collatz and Sinogowits [14], who proved that λ 1 , the greatest eigenvalue of the adjacency matrix, satisfies the inequality λ 1 ≥ 2m/n.In the case of connected graphs, equality occurs if and only if this graph is regular.Thus, the Collatz-Sinogowitz irregularity measure is Its application is not easy, because the λ 1 -value of a (non-regular) graph G cannot be directly deduced from the structure of G (see, for instance, [45,48]).
Certain irregularity measures depend solely on the degree sequence d, and can thus be written as Irr = f (d).These automatically imply that two graphs with equal degree sequence are equally irregular.In particular, according to such irregularity measures, the graphs G 3 and G 4 in Fig. 2 would be claimed to be equally irregular.
The simplest and best known index of this kind is that of Bell [10].He measured the irregularity of a graph by means of the variance of its vertex degrees.Bell's irregularity index is recalling that the average value of vertex degrees is 2m/n.
Another direction of approaching the irregularity is by taking into account the difference of degrees of adjacent vertices.By this, two graphs with equal degree sequence need not necessarily be considered as equally irregular.In particular, the irregularity of the graphs G 3 and G 4 from Fig. 2 will be found to differ.
Accordingly, Albertson conceived the irregularity of a graph G as [7] Irr which is usually referred to as the Albertson index [22,24,38,44,49,50], although the name "third Zagreb index" has also been proposed [21].A similar, yet not much studied quantity would be [23] Irr Recently [1,3], the total irregularity was introduced, defined as which in analogy with (3) could be modified as Some of the above specified irregularity measures are related to the two Zagreb indices M 1 (G) and M 2 (G), and the F -index [23,25,27,28,30,47] Namely, For comparative studies of Irr CS , Irr Alb , Irr tot , and Irr Bell see [10,15,17,31].

Concluding remarks
The concept of "graph irregularity", i.e., the extent by which a graph deviates from "regularity" is a notion that has a vague meaning.From our everyday's experience we have some "intuitive" feeling about this notion, and in some cases the majority of us would agree that a particular graph is more "irregular" than another.(For examples, see Fig. 2.) When one attempts to quantify this concept, and give a precise method for its measuring, then we encounter ambiguity.Simply: measures of "graph irregularity" can be constructed in numerous agreeable, but different, ways.
In this survey we presented a dozen of such measures, whereas a skilled scholar could design a dozen more.
Needless to say that this arbitrariness is an opportunity to produce a multitude papers that resemble serious scientific research.

Fig. 2 .
Fig. 2. Two pairs of graphs illustrating the weakness of the concept of irregularity index Irr d .For details see text.