Ordering graphs with large eccentricity-based topological indices

*Correspondence: qixuli-1212@163.com 2School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China 3Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303, USA Full list of author information is available at the end of the article Abstract For a connected graph, the first Zagreb eccentricity index ξ1 is defined as the sum of the squares of the eccentricities of all vertices, and the second Zagreb eccentricity index ξ2 is defined as the sum of the products of the eccentricities of pairs of adjacent vertices. In this paper, we mainly present a different and universal approach to determine the upper bounds respectively on the Zagreb eccentricity indices of trees, unicyclic graphs and bicyclic graphs, and characterize these corresponding extremal graphs, which extend the ordering results of trees, unicyclic graphs and bicyclic graphs in (Du et al. in Croat. Chem. Acta 85:359–362, 2012; Qi et al. in Discrete Appl. Math. 233:166–174, 2017; Li and Zhang in Appl. Math. Comput. 352:180–187, 2019). Specifically, we determine the n-vertex trees with the i-th largest indices ξ1 and ξ2 for i up to n/2 + 1 compared with the first three largest results of ξ1 and ξ2 in (Du et al. in Croat. Chem. Acta 85:359–362, 2012), the n-vertex unicyclic graphs with respectively the i-th and the j-th largest indices ξ1 and ξ2 for i up to n/2 – 1 and j up to 2n/5 + 1 compared with respectively the first two and the first three largest results of ξ1 and ξ2 in (Qi et al. in Discrete Appl. Math. 233:166–174, 2017), and the n-vertex bicyclic graphs with respectively the i-th and the j-th largest indices ξ1 and ξ2 for i up to n/2 – 2 and j up to 2n/15 + 1 compared with the first two largest results of ξ2 in (Li and Zhang in Appl. Math. Comput. 352:180–187, 2019), where n≥ 6. More importantly, we propose two kinds of index functions for the eccentricity-based topological indices, which can yield more general extremal results simultaneously for some classes of indices. As applications, we obtain and extend some ordering results about the average eccentricity of bicyclic graphs, and the eccentric connectivity index of trees, unicyclic graphs and bicyclic graphs.


Introduction
Topological indices are numerical graph invariants that quantitatively characterize molecular structure, which are useful molecular descriptors that found considerable use in QSPR and QSAR studies [20,21]. Several graph invariants based on vertex eccentricities have attracted much attention and have been subject to a large number of studies. We mainly study two kinds of eccentricity-based topological indices, that is, the first Zagreb eccentricity index and the second Zagreb eccentricity index, special cases of which have been studied due to their predictive capabilities for the physical and chemical properties of molecules.
All graphs considered in this paper are finite, simple, and connected. Let G be a graph with a vertex set V (G) and an edge set E(G). For u ∈ V (G), e G (u) denotes the eccentricity of u in G, which is equal to the largest distance from u to other vertices of G.
The first Zagreb eccentricity index of G is defined as while the second Zagreb eccentricity index of G is defined as These two types of Zagreb eccentricity indices were introduced by Vukičević and Graovac [22]. Some mathematical and computational properties of the Zagreb eccentricity indices have been obtained in [4,6,12,13,15,16,24]. Du et al. [6] determined the n-vertex trees with maximum, second-maximum, and third-maximum Zagreb eccentricity indices. Qi and Du [15] determined the trees with minimum Zagreb eccentricity indices when domination number, maximum degree, and bipartition size are respectively given, and they discussed the trees with maximum Zagreb eccentricity indices when domination number is given. Qi et al. [16] determined the n-vertex unicyclic graphs with maximum, secondmaximum eccentricity index ξ 1 , and maximum, second-maximum, and third-maximum eccentricity index ξ 2 . Li and Zhang [12] determined the n-vertex bicyclic graphs with maximum and second-maximum eccentricity index ξ 2 . For a connected graph, the first Zagreb eccentricity index ξ 1 is defined as the sum of the squares of the eccentricities of all vertices, and the second Zagreb eccentricity index ξ 2 is defined as the sum of the products of the eccentricities of pairs of adjacent vertices. In this paper, we mainly present a different and universal approach to determine the upper bounds respectively on the Zagreb eccentricity indices of trees, unicyclic graphs and bicyclic graphs, and characterize these corresponding extremal graphs, which extend the ordering results of trees, unicyclic graphs, and bicyclic graphs in [6,12,16]. Specifically, we determine the n-vertex trees with the i-th largest indices ξ 1 and ξ 2 for i up to n/2+1 compared with the first three largest results of ξ 1 and ξ 2 in [6], the n-vertex unicyclic graphs with respectively the i-th and the j-th largest indices ξ 1 and ξ 2 for i up to n/2 -1 and j up to 2n/5 + 1 compared with respectively the first two and the first three largest results of ξ 1 and ξ 2 in [16], and the n-vertex bicyclic graphs with respectively the i-th and the j-th largest indices ξ 1 and ξ 2 for i up to n/2 -2 and j up to 2n/15 + 1 compared with the first two largest results of ξ 2 in [12], where n ≥ 6. More importantly, we propose two kinds of index functions for the eccentricity-based topological indices, which can yield more general extremal results simultaneously for some classes of indices. As applications, we obtain and extend some ordering results about the average eccentricity of bicyclic graphs and the eccentric connectivity index of trees, unicyclic graphs and bicyclic graphs.

Preliminaries
Let n 0 and d 0 be positive integers. Let T n ≥n 0 (res. U n ≥n 0 , B n ≥n 0 ) be the set of n-vertex trees (res. unicyclic graphs, bicyclic graphs), where n ≥ n 0 . Let T(n ≥n 0 , d ≤d 0 ) (res. U(n ≥n 0 , d ≤d 0 ), Figure 1 The graph T i n B(n ≥n 0 , d ≤d 0 )) be the set of n-vertex trees (res. unicyclic graphs, bicyclic graphs) with the diameter d, where n ≥ n 0 and d ≤ d 0 .
Let P n be the path on n vertices. For 2 ≤ d ≤ n -1, let T (n,d) = {T a n,d : 1 ≤ a ≤ (n + 1d)/2 }, where T a n,d is the n-vertex tree obtained by attaching a and n + 1ad pendent vertices respectively to the two end vertices of the path P d-1 . For n ≥ 4, let T i n be the tree formed by attaching a pendent vertex v n-1 to a vertex v i of the path P n- Fig. 1). In particular, T 0 n = P n . The following observation is obvious.

Observation 2.1
If G is a connected graph such that Ge is also connected for e ∈ E(G), then e G (u) ≤ e G-e (u) for u ∈ V (G), and thus ξ 1 (G) ≤ ξ 1 (Ge).
with equality if and only if G ∈ T (n,d) , where if d is even, if d is odd, and f 1 (n, d) and f 2 (n, d) are increasing for 2 ≤ d ≤ n -1.

Figure 2
The graph P n,3 (i) Figure 3 The graph P n,4 (i) 2 ) can be obtained from T i+1 n by adding an edge v i v n-1 (res. v i v n-1 , two edges v i v n-1 and v i-1 v n-1 ). From the above fact we may easily find that vertices in P n,3 (i), P n,4 (i -1), B n (i -1), and T i+1 n with the same label have equal eccentricity for 1 ≤ i ≤ n-4 2 , and so do the vertices in P n,3 (0) and T 1 n . Thus result (1) follows.

Ordering trees with large Zagreb eccentricity indices
Theorem 3.1 Among the graphs in T n ≥3 , P n for n ≥ 3 is the unique graph with the largest eccentricity indices ξ 1 and ξ 2 , and T i n for n ≥ 4 and 1 ≤ i ≤ n/2 -1 is the unique graph with the (i + 1)th largest eccentricity indices ξ 1 and ξ 2 , where if n is even, if n is odd, if n is even, if n is odd, 12 12 + (ni -1) 2 if n is even, 7n 3 -30n 2 +41n- 18 12 + (ni -1) 2 if n is odd, Thus T i n is the unique n-vertex tree with the (i + 1)th largest eccentricity indices ξ 1 and ξ 2 , where 1 ≤ i ≤ n/2 -1 . The result follows.
In fact, from the proof of Theorem 3.1, we have the following corollary easily.
Proof Let G ∈ U(n ≥6 , d ≤n-2 ). From Lemmas 2.3(2) and 3.3, we only need to show that ξ 1 (G) < ξ 1 (P n,3 ( n-4 2 )) for G ∈ U(n ≥6 , d ≤n-3 ). It is easy to find that there exists an edge e on the cycle of G such that the diameter of Ge is at most n -3. Note that f 1 (n, n -3) < ξ 1 (T Thus we have the result.

Ordering bicyclic graphs with large Zagreb eccentricity indices
Proof If n is even, then G = B n ( n- 4 2 ). If n is odd, then G = B n ( n-5 2 ). By direct computation, we have Thus the result follows.

Ordering graphs with large eccentricity-based topological indices
Most papers on this topic study just one topological index and find its extremal values (or perhaps several near-extreme values) over n-vertex trees or other simple classes. We propose studying such problems in terms of general properties of some index functions for the eccentricity-based topological indices. Requiring only the properties needed for the argument yields a more general extremal result simultaneously for a class of indices.
In the following, we consider two kinds of index functions, that is, a vertex-weight index function and an edge-weight index function.

Definition 4.1
The weight ω(u) of a vertex u in a graph G is e G (u). Given a positive realvalued function t v , the vertex-weight index function for a graph G is defined by The total eccentricity index of G, introduced by Farooq et al. [8], is defined as τ (G) = u∈V (G) e G (u) using t v (ω(u)) = ω(u); the average eccentricity of G, introduced by Bukley et al. [1], is defined as avec(G) = 1 n τ (G) using t v (ω(u)) = 1 n ω(u). For more recent results on average eccentricity, see [2,3,5,11].
Note that those above indices have similar extremal values (or perhaps several nearextremal values) over n-vertex trees, unicyclic graphs, and bicyclic graphs. Tang and Zhou have determined similar extremal values of the average eccentricity over n-vertex trees [18] and unicyclic graphs [19]. Like the discussion of the first Zagreb eccentricity index, we also obtain a similar result about the average eccentricity of bicyclic graphs in the following Theorem 4.2, whose proof is omitted since we use a similar method. Theorem 4.2 Among the graphs in B n ≥6 , B n (i) is the unique graph with the (i + 1)th largest average eccentricity, equal to 3(n-1) 2 +2n-4i-11 4n for even n and 3(n-1) 2 +2n-4i-10 4n for odd n, where 0 ≤ i ≤ n 2 -3.

Definition 4.3
The weight ω(e) of an edge e = uv in a graph G is e G (u)e G (v), and the weight ω * (e) of an edge e = uv in a graph G is e G (u) + e G (v). Given a positive real-valued function t e , the edge-weight index functions for a graph G are defined by f 1 (G; e) = e∈E(G) t e (ω(e)), f 2 (G; e) = e∈E(G) t e (ω * (e)), and f 3 (G; e) = e∈E(G) t e (ω(e), ω * (e)), respectively.
The second Zagreb eccentricity index of G is defined as ξ 2 (G) = uv∈E(G) e G (u)e G (v) using t e (ω(e)) = ω(e). The eccentric connectivity index of G, introduced by Sharma et al. in [17], is defined as ξ c (G) = v∈V (G) e G (v)d G (v) = uv∈E(G) (e G (u) + e G (v)) using t e (ω * (e)) = ω * (e), which is also called the third Zagreb eccentricity index by Ghorbani et al. [10]. For some recent results of the eccentric connectivity index, see [14,23].