Zagreb Eccentricity Indices of the Generalized Hierarchical Product Graphs and Their Applications

A topological index is a real number associatedwith chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity, or biological activity, which is used to understand properties of chemical compounds in theoretical chemistry [1]. Up to now, hundreds of topological indices have been defined in chemical literatures, various applications of these topological indices have been found, andmanymathematical properties are also investigated. Wiener index W is the first topological index, introduced by American chemist Wiener, for investigating boiling points of alkanes in 1947 [2]. The well known degree-based topological indices are the first and second Zagreb indices M 1 and M 2 , which have been introduced byGutman and Trinajstić [3] and applied to study molecular chirality in quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR) analysis and so forth. Resently, the first and second Zagreb eccentricity indices M∗ 1 and M∗ 2 have been introduced byGhorbani andHosseinzadeh [4] andVukičević and Graovac [5] as the revised version of the Zagreb indices


Introduction
A topological index is a real number associated with chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity, or biological activity, which is used to understand properties of chemical compounds in theoretical chemistry [1].
Up to now, hundreds of topological indices have been defined in chemical literatures, various applications of these topological indices have been found, and many mathematical properties are also investigated.Wiener index  is the first topological index, introduced by American chemist Wiener, for investigating boiling points of alkanes in 1947 [2].The well known degree-based topological indices are the first and second Zagreb indices  1 and  2 , which have been introduced by Gutman and Trinajstić [3] and applied to study molecular chirality in quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR) analysis and so forth.Resently, the first and second Zagreb eccentricity indices  * 1 and  * 2 have been introduced by Ghorbani and Hosseinzadeh [4] and Vukičević and Graovac [5] as the revised version of the Zagreb indices  1 and  2 , respectively.They computed the Zagreb eccentricity indices of some composite graphs and showed that  * 1 ()/|| ≥  * 2 ()/|()| holds for all acyclic and unicyclic graphs and that neither this nor the opposite inequality holds for all bicyclic graphs.For further results of the Zagreb eccentricity indices, we encourage the reader to refer to [6][7][8].
In 2009, Spain mathematicians Barrière and coauthors [9] introduced a new composite graph, namely, hierarchical product graph.In the same year, this team also reported a generalization of both Cartesian and the hierarchical product of graphs, namely, the generalized hierarchical product of graphs in [10].After that, many results for some topological indices of the (generalized) hierarchical product of graphs are reported; see [11][12][13][14][15][16][17].
In this paper, the Zagreb eccentricity indices of the generalized hierarchical product graph () ⊓  are computed and as some special cases of ()⊓, the Zagreb eccentricity indices of the Cartesian product graph ◻, the -sum graph  +  , and the cluster product graph {} are determined, respectively.Moreover, as applications, we present explicit formulas for the  * 1 and  * 2 indices of the  4 nanotorus   ◻  , the  4 nanotubes   ◻  , the zig-zay polyhex nanotube  6 [2, 2], the hexagonal chain   , and so forth.

Preliminaries
Throughout this paper, all graphs are simple, finite, and undirected.For terminology and notations that are not defined here, we refer the reader to West [18].
Let  = ((), (),   ) be a graph with the vertex set () ̸ = 0, the edge set (), and an incidence function   that associates with each edge of , an unordered pair of vertices of .If  is an edge and  and V are vertices such that   () = V, then  is said to join  and V, and the vertices  and V are called the ends of .The cardinality of () and () is denoted by || and |()|, respectively.We denote the degree and the neighborhood of a vertex V of  by   (V) and   (V); then   (V) = |  (V)|.As usual, the distance between vertices  and V of a connected graph , denoted by   (, V), is defined as the number of edges in a shortest path connecting the vertices  and V. Suppose that ( | ) = ∑ V∈()   (, V) and  2 ( | ) = ∑ V∈() (  (, V)) 2 .The eccentricity   (V) of a vertex V in  is the largest distance between V and any other vertex  of ; that is,   (V) = max ∈()   (, V).For two graphs  and , if there exist two bijections  : () → () and  : () → () such that   () = V if and only if   (()) = ()(V), then we say that  and  are isomorphic, denoted by  ≅ .Let Top() denote a certain topological index of .In general, if  ≅ , then Top() = Top().

Zagreb Eccentricity Indices of Generalized Hierarchical Product Graphs
In this section, we calculate the Zagreb eccentricity indices of the generalized hierarchical product graphs.
(i) By the definition of Zagreb eccentricity index  * 1 and Lemma 6, we have (ii) We partition the edges of () ⊓  into two subsets  1 and  2 , as follows: From the definition of Zagreb eccentricity index  * 2 and Lemma 6, we get This completes the proofs.
Lemma 12 (see [19]).Let ◻  =1   be Cartesian product of  ≥ 2 connected graphs   .Then Corollary 13.Let ◻  =1   be Cartesian product of  ≥ 2 simple connected graphs   .Then Proof.The case  = 2 is proved in Corollary 9. We prove the assertion by induction.Suppose the result is valid for  graphs.Then by Lemma 12, we have By induction, we can easily prove that Therefore, by Corollary 9, Lemma 12 and the formula as above, using a similar method of proof in Corollary 13, we can obtain Corollary 16.

Zagreb Eccentricity Indices of 𝑆-Sum Graphs
Let  be a connected graph.The vertices of a Line graph () are the edges of .Two edges of  that share a vertex are considered to be adjacent in ().A Subdivision graph () is the graph obtained by inserting an additional vertex in each edge of .That is, each edge of  is replaced by a path of length two.

Zagreb Eccentricity Indices of Cluster and Corona Product Graphs
The cluster product, corona product, and join of two graphs are important graph operations defined as below.
Definition 26 (see [20]).The cluster product graph {} is obtained by taking one copy of  and || copies of a rooted graph  and by identifying the root of the th copy of  with the th vertex of ,  = 1, 2, . . ., ||.
Definition 27 (see [20]).The corona product graph  ⊙  is obtained by taking one copy of  and || copies of  and by joining each vertex of the th copy of  to the th vertex of ,  = 1, 2, . . ., ||.
Let  and  be two simple graphs.If || =  and |()| = , then we say that  is an (, )-graph.According to the definitions of the cluster and corona products, if  is an (, )graph and  is a (  ,   )-graph, then {} is an (  , +  )graph and  ⊙  is an (  + ,  +   +   )-graph.30) and (31), using the same method as above, the corresponding equations ( 34) and ( 35) are also obtained, respectively.