On the construction and comparison of graph irregularity indices

Abstract

ABSTRACT. Irregularity indices are generally used for quantitative characterization of topological structure of non-regular graphs. According to a widely accepted preconception, using a topological invariant (called a graph irregularity index) for that purpose, the results of graph irregularity classification should be consistent with our subjective judgements (intuitive feeling). In the case of structurally strongly similar graphs, it is difficult to select the proper irregularity index by which the irregularity ranking (ordering) of graphs can be performed in a consistent manner to our preliminary expectations. In this work we investigate various possibilities of constructing so-called composite irregularity indices obtained as a function of traditional irregularity indices and test their discriminatory performance. Moreover, it has been demonstrated in examples, that in some cases the subjective evaluation of graph irregularities leads to false conclusions. This phenomenon is called the paradox of quantitative graph irregularity characterization. From our study we have concluded that results of graph irregularity measuring depend not only on the choice of irregularity indices but it is influenced strongly on the preselected set of graphs to be investigated. Similar problems arise for the quantitative evaluation of information content, complexity or branching of molecular graphs. 54

Theoretical Analysis of Superphenalene Using Different Kinds of VDB Indices

“…Let Γ � SP(n) be the molecular graph of a superphenalene. e cardinality of superphenalene is V(Γ) � 54n 2 − 72n + 24 and E(Γ) � 81n 2 − 120n + 45. is theorem is proved in a similar way to that of in [38]. In order to prove the above results, the definitions of VDB indices, VDB entropy measures and the edge partition Table 1 have been used.…”

Some generalizations of the total irregularity of graphs

“…An irregularity measure (IM) of a connected graph G is a non-negative graph invariant satisfying the property: IM(G) = 0 if and only if G is regular. There exist several degreebased and eigenvalue-based graph irregularity measures [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]. The following irregularity measure, defined for a connected (n, m)-graph G, is called Collatz-Sinogowitz irregularity index [6]:…”

“…It should be noted that in the formula given in Lemma 4.3, ∑ n i=1 d p i is a special case of the general zeroth-order Randić index [18].…”

On Bond-Additive and Atoms-Pair-Additive Indices of Graphs

“…A topological index T I is said to be a graph irregularity index (or irregularity measure) if T I(G) ≥ 0 and if the equality T I(G) = 0 holds if and only if graph G is regular (see [1,2,6,11,31,38,47,[54][55][56]). For a graph G, its topological indices of the form uv∈E(G) β(u, v) are known as bond-additive indices [67], where β is a real-valued function satisfying the condition β(u, v) = β(v, u).…”