Reciprocal degree distance of Eliasi-Taeri sums of graphs

For a connected graph G , the reciprocal degree distance is deﬁned as RDD ( G ) = (cid:80) u,v ∈ V ( G ) (cid:0) d G ( u ) + d G ( v ) (cid:1)(cid:0) 2 d G ( u, v ) (cid:1) − 1 , where d G ( u ) denotes the degree of a vertex u in G and d G ( u, v ) represents the distance between the vertices u and v in G . In this paper, upper bounds for the reciprocal degree distance of graphs, arising from four operations introduced by Eliasi and Taeri in [ Discrete Appl. Math. 157 (2009) 794–803], are provided.


Introduction
Throughout this paper, we assume that G is a finite, connected, undirected, and simple graph, with vertex set V (G) and edge set E(G). The degree of a vertex x in G is denoted by d G (x) and the distance between u, v ∈ V (G) is denoted by Topological indices provide contemporary tools to predict diverse psychochemical features of chemical compounds. A lot of available research shows that topological indices are easy to compute and they efficiently encrypt significant structural information of chemical compounds. Distance-based topological indices form a vital class of indices and these indices have significantly better efficiency.
The Zagreb indices have been introduced around a half century ago by Gutman and Trinajestić in [6]. They are defined as and In the definitions of the Zagreb indices, if one takes the sums over the edges of the complement of G then the resulting quantities are the Zagreb coindices. More precisely, the first and second Zagreb coindices of G are defined as and One of the oldest and well-studied topological indices is the Wiener index. The Wiener index of G is defined as the sum of the distances of all unordered vertex pairs in G, that is, Another distance-based topological index, defined in a fully analogous manner to the Wiener index, is the Harary index which is equal to the sum of the the reciprocal distances overall all unordered vertex pairs in G, that is, .
Dobrynin and Kochetova [3], and Gutman [5] independently proposed a vertex-degree-weighted version of the Wiener index known as the degree distance (DD), which is defined as The reciprocal degree distance (RDD) of a connected graph G is defined [1] as One can observe that the degree distance and reciprocal degree distance are defined by using the concepts of the first Zagreb index, Wiener index, and Harary index. Chemical applications and mathematical properties of the reciprocal degree distance are well studied [1,8,12]. Hua and Zhang [7] obtained lower and upper bounds for the reciprocal degree distance of graphs in terms of other graph parameters. The mathematical behavior of the reciprocal degree distance of some composite graphs was analysed in [10,11]. In the present paper, we obtain upper bounds for the reciprocal degree distance of F -sums of graphs introduced in [4].

Main results
In order to state and prove the main results, we need some definitions. In the following, we define four graphs generated from a given connected graph G.
(i). The graph formed by inserting a new vertex on every edge of G is known as the subdivision graph S(G) of G.
(ii). The graph deduced from G by first inserting an additional vertex corresponding to every edge of G and then joining every newly added vertex to the end vertices of the corresponding edge is denoted by R(G).
(iii). The graph deduced from G by first inserting an additional vertex into every edge of G and then joining with edges the pairs of new vertices on adjacent edges of G is denoted by Q(G).
(iv). The total graph T (G) of G has the vertex set consisting of the vertices and edges of G. Two vertices in T (G) are adjacent if and only if they are either adjacent or incident in G.
For F ∈ {S, R, Q, T }, the F -sum or Eliasi-Taeri sum of the graphs G 1 (V 1 , E 1 ) and G 2 (V 2 , E 2 ) is denoted by G 1 + F G 2 and is defined as the graph with the vertex set V (G 1 + F G 2 ) = (V 1 ∪ E 1 ) × V 2 and two vertices (u 1 , u 2 ) and (v 1 , v 2 ) of G 1 + F G 2 are adjacent if and only if either u 1 = v 1 ∈ V 1 and u 2 v 2 ∈ E 2 or u 2 = v 2 and u 1 v 1 ∈ E(F (G 1 )).
The following known lemmas are used in the proofs of our main results.
Lemma 2.7 (see [9]). Let g be a convex function on the interval Z and z 1 , z 2 , . . . , z n ∈ Z. Then Now, we are able to state and prove the first main result of this paper.
By the definition of the reciprocal degree distance, we have where S 1 , S 2 , S 3 , S 4 are the sums in order.
First, we compute S 1 . By Lemma 2.3, we have .
with equality if and only if d F (G1) (u 1 , v 1 ) = d G2 (u 2 , v 2 ). Therefore, By the definitions of the Harary index and reciprocal degree distance, we obtain By Lemma 2.3, we obtain .
By Jensen's inequality, we have, In order to calculate S 3 and S 4 , we assume that u and v are the vertices such that u ∈ E(G 1 )×V (G 2 ) and v ∈ E(G 1 )×V (G 2 ). By Lemma 2.4, we have By using Lemma 2.4, we have .
By Jensen's inequality, we have with equality if and only if d F (G1) (u 1 , v 1 ) = d G2 (u 2 , v 2 ). Thus, From the obtained values and inequalities of S 1 , S 2 , S 3 , and S 4 , we get Now, we calculate the sums given in (1) separately.
(i). By Lemma 2.6, one has (ii). By the definition of the Harary index, we get (iii). By Lemma 2.6, we have Now, by making use of (i)-(vi) in (1), we get Next, we state and prove the second main result of this paper.

Theorem 2.2.
Suppose that G i is a connected graph with n i vertices and m i edges, i = 1, 2, and take F ∈ {Q, T }. Then By the definition of the RDD index, we have where S 1 , S 2 , · · · , S 5 are the sums in order. First, we compute S 1 and S 2 . By Lemma 2.4 and a similar way as used in Theorem 2.1, we obtain Similarly, we have To calculate S 3 and S 4 , we assume that u and v are the vertices such that u ∈ E(G 1 ) × V (G 2 ) and v ∈ E(G 1 ) × V (G 2 ). By Lemma 2.4, we have By Lemma 2.5, we get Finally, the quantity of S 5 is computed as follows. .
We end this paper by posing the following open problem which is related to our main results.