Sharp Bounds for the General Sum-Connectivity Indices of Transformation Graphs

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Introduction
In this paper, we consider simple, undirected, and connected graphs.Let  be the graph with vertex set () and edge set ().The order and size of  are denoted by  and , respectively.For a vertex  ∈ (),   () denotes the degree of .Two vertices in  are adjacent if and only if they are end vertices of an edge, and each of the two vertices is called incident to the edge.Besides, two edges are adjacent to each other if and only if they share a common vertex.The minimum and maximum degrees of graph  are denoted by () and Δ(), respectively.We will use the notations   ,   , and   for a path, cycle, and complete graph of order  [1], respectively.
The complement of , denoted by , is the graph with () = () and two vertices in  are adjacent if and only if they are not adjacent in .Thus, the size of  is (  2 ) −  and if  ∈ () then   () =  − 1 −   ().
A topological index is a numeric quantity associated with a graph which characterizes the topology of graph.A topological index Top() of a graph  is equal to the topological index Top() of , if and only if two graphs  and  are isomorphic.The idea of topological index appears from work done by Wiener in 1947, this index is called Wiener index.The first and second Zagreb indices have been introduced by Gutman and Trinajestić [2].These indices are defined on the ground of vertex degrees as follows: ()   () . ( The Randić connectivity index was defined in 1975 by Randić [3].It has been extended to the general Randić connectivity index.The general Randić connectivity index (general product-connectivity index) was defined by Bollobás and Erdős [4] as follows: where  is a real number.Then  −1/2 is the classical Randić connectivity index.The sum-connectivity index was proposed in [5].This concept was extended to the general sumconnectivity index in [6], which is defined as where  is a real number.Then  −1/2 () is the classical sum-connectivity index.The sum-connectivity index and the product-connectivity index correlate well with the -electron energy of benzenoid hydrocarbons [7].The total graph () of the graph  is a graph whose vertex set is the union of () and () such that  ∈ (()) if and only if  and  are either adjacent or incident in  [8].Let , V, and  be the variables having values + or −.The transformation graph  V is a graph whose vertex set is the union of () and (), and  ∈ ( V ) if and only if (1) ,  ∈ (); then  = + or  = − if  and  are adjacent or nonadjacent in , respectively; (2) ,  ∈ (); then V = + or V = − if  and  are adjacent or nonadjacent in , respectively; (3)  ∈ () and  ∈ (); then  = + or  = − if  and  are incident or nonincident in , respectively.
The concepts of semitotal-point graph and semitotal-line graph are introduced by Sampathkumar and Chikkodimath [9].The semitotal-point graph  1 () is a graph whose vertex set is the union of () and (), and  ∈ ( 1 ()) if and only if (i)  and  are adjacent vertices in  or (ii) one is a vertex of  and the other is an edge of  incident to it.Thus, semitotal-point graph has  +  number of vertices and 3 number of edges.
The semitotal-line graph  2 () is a graph whose vertex set is the union of () and (), and  ∈ ( 2 ()) if and only if (i)  and  are adjacent edges in  and (ii) one is a vertex of  and the other is an edge of  incident to it.Thus, semitotalline graph has  +  number of vertices and (1/2) 1 () +  number of edges.
Eventually, many properties of these transformation graphs can be determined.For example, the Zagreb indices of transformation graphs and total transformation graphs were calculated by Basavanagoud and Patil [10] and Hosamani and Gutman [11], respectively.Wu and Meng [12] investigated the basic properties (connectedness, graph equations and iteration, and diameter) of total transformation.Xu and Wu [13] determined the connectivity, the Hamiltonian, and the independence number of  −+− .Yi and Wu [14] determined the connectivity, the Hamiltonian, and the independence number of  ++− .
In this paper, we obtain lower and upper bounds for the general sum-connectivity indices of the above-defined transformation graphs.

Main Results
In this section, we discuss the lower and upper bounds for the general sum-connectivity indices of transformation graphs defined in Section 1. where the equalities hold if and only if  is a regular graph.
Similarly, we calculate ( If  is a regular graph then we obtain the equalities in (28) and (29).
In fully analogous manner, we also arrive at the following.