Harary-Albertson index of graphs

where d(u) and d(u, v) are the degree of the vertex u and the distance between the vertices u and v in G, respectively. This new index is useful in predicting physico-chemical properties with high accuracy compared to some classic topological indices. Mathematical relations between the Harary-Albertson index and other classic topological indices are established. The extremal values of the Harary-Albertson index for trees of given order are also determined.


Introduction
Let G be a simple undirected connected graph with the vertex set V (G) and edge set E(G). For v ∈ V (G), d(v) = d G (v) denotes the degree of vertex v in G. The minimum and the maximum degree of G are denoted by δ(G) and ∆(G), or simply δ and ∆, respectively. A pendant vertex of G is a vertex of degree one. The distance between two vertices u, v ∈ V (G), denoted by d(u, v) = d G (u, v), is defined as the length of a shortest path between u and v. The eccentricity of v, ε(v), is the distance between v and any vertex which is furthest from v in G. The diameter of G is the maximum eccentricity in G, denoted by D(G). Similarly, the radius of G is the minimum eccentricity in G, denoted by r(G). The join G 1 ∨ G 2 of the graphs G 1 and G 2 is obtained from G 1 ∪ G 2 by adding to it all edges between vertices from V (G 1 ) and V (G 2 ). The lexicographic product of the graphs G 1 and G 2 is denoted by G 1 [G 2 ], and it is the graph with vertex set V (G 1 ) × V (G 2 ), and two vertices (u 1 , v 1 ) and (u 2 , v 2 ) are adjacent if (u 1 is adjacent to u 2 in G 1 ) or (u 1 = u 2 and v 1 and v 2 are adjacent in G 2 ). Denote by P n , S n and G the path, the star and the complement of G, respectively.
In recent decades, the topological indices (graphical invariants or topological molecular descriptors) have been extensively studied in various areas of mathematics [8,11], physics [12], informatics [16], biology [3], especially in chemical disciplines [4,5,14,17], such as chemical documentation, isomer discrimination, study of molecular complexity, chirality, similarity/dissimilarity, QSAR/QSPR, drug design, database selection, lead optimization, etc. In particular, the following classic topological indices appear more frequently in the literature in the above related fields.
• second Zagreb index [9]: • total irregularity index [1]: • atom-bond-connectivity index [6]: • Harary index [10,13]: Based on the Albertson index, Abdo, Brandt and Dimitrov [1] proposed the total irregularity index, which can be applied as irregularity measure when the adjacency information of the vertices is unknown, because the total irregularity index of a graph depends only on its degree sequence, see for example Figure 1. However, it is well known that there are many graphs with the same degree sequence, which makes the structure discriminating ability of the total irregularity index lower. Usually, distance-based topological indices have better structure discriminating ability than degree-based topological indices. Moreover, any interaction must decrease with the increase of the distance between the interacting particles in general situation. In order to improve the structure discriminating ability and maintain certain irregularity measuring ability, we propose the Harary-Albertson index of a connected graph G as follows: In this paper, we confirm the suitability of the Harary-Albertson index in quantitative structure-property relationship (QSPR) analysis by the correlation of acentric factor (AcenFac), entropy (S), SNar and HNar with the Harary-Albertson index for octane isomers. Secondly, we obtain mathematical relations between the Harary-Albertson index and other classic topological indices. Finally, we study the extremal values of the Harary-Albertson index for trees of given order.

The Harary-Albertson index in QSPR analysis
In this section, the chemical applicability of the Harary-Albertson index is investigated. We obtain the data related to octanes, listed in Table 1, using matlab software and the experimental data set octane isomers. We get that the correlation coefficient of the Harary-Albertson index with AcenFac, S, SNar and HNar are −0.9667, −0.9073, −0.9786 and −0.9820, respectively, for octane isomers. Thus the Harary-Albertson index can help to predict these physico-chemical properties of octane isomers. These results confirm the suitability of the Harary-Albertson index in QSPR analysis. Meanwhile, the following equations give the regression models for the Harary-Albertson index. In order to prove that the Harary-Albertson index shows better predictive capability, we study the correlation of some classic topological indices like the second Zagreb index, Sombor index, Randić index, Albertson index, total irregularity index, atom-bond-connectivity index, Wiener index, Harary index with AcenFac, S, SNar and HNar, shown in Table 2. It is not difficult to find that sometimes the Harary-Albertson index shows better predictive capability than the existing indices.

Harary-Albertson and other degree-based indices
Proof. By the definition of HA(G), we have the proof.

Theorem 3.2. Let G be a triangle-and quadrangle-free irregular connected graph with n ≥ 4 vertices and m edges. Then
Proof. Let d(G, k) be the number of vertex pairs of the graph G that are at distance k. If G be a triangle-and quadranglefree irregular connected graph with n ≥ 4 vertices and m edges, from [20], we have By the definition of HA(G), we have A graph is said to be self-complementary if it is isomorphic with its complement. Clearly, if G is a self-complementary graph, then HA(G) = HA(G).

Theorem 3.3. Let G be an irregular connected graph with n vertices and maximum degree ∆(G) ≤ n − 2. Then
In particular, if D(G) ≥ 3, then Furthermore, if D(G) ≥ 4, then Since G is a connected graph with maximum degree ∆(G) ≤ n − 2, we know that G is a connected graph with r(G) ≥ 2. By the definition of the Harary-Albertson index, we have If D(G) ≥ 3, then D(G) ≤ 3. Thus, we have If D(G) ≥ 4, then D(G) = 2. By a similar reasoning as the above, we have the proof.