Given a graph G, the general sum-connectivity index is defined as χα(G)=∑uv∈E(G)dGu+dGvα, where dG(u) (or dG(v)) denotes the degree of vertex u (or v) in the graph G and α is a real number. In this paper, we obtain the sharp bounds for general sum-connectivity indices of several graph transformations, including the semitotal-point graph, semitotal-line graph, total graph, and eight distinct transformation graphs Guvw, where u,v,w∈+,-.

Fundamental Research Funds for the Central Universities26520151932652017146National Natural Science Foundation of China117015301. Introduction

In this paper, we consider simple, undirected, and connected graphs. Let G be the graph with vertex set V(G) and edge set E(G). The order and size of G are denoted by n and e, respectively. For a vertex a∈V(G), dG(a) denotes the degree of a. Two vertices in G 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 G are denoted by δ(G) and Δ(G), respectively. We will use the notations Pn, Cn, and Kn for a path, cycle, and complete graph of order n [1], respectively.

The complement of G, denoted by G¯, is the graph with V(G¯)=V(G) and two vertices in G¯ are adjacent if and only if they are not adjacent in G. Thus, the size of G¯ is n2-e and if a∈V(G¯) then dG¯(a)=n-1-dG(a).

A topological index is a numeric quantity associated with a graph which characterizes the topology of graph. A topological index Top(G) of a graph G is equal to the topological index Top(H) of H, if and only if two graphs G and H 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:(1)M1G=∑a∈VGdGa2,M2G=∑ab∈EGdGadGb.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:(2)RαG=∑ab∈EGdGadGbα,where α is a real number. Then R-1/2 is the classical Randić connectivity index. The sum-connectivity index was proposed in [5]. This concept was extended to the general sum-connectivity index in [6], which is defined as(3)χαG=∑ab∈EGdGa+dGbα,where α is a real number. Then χ-1/2(G) is the classical sum-connectivity index. The sum-connectivity index and the product-connectivity index correlate well with the π-electron energy of benzenoid hydrocarbons [7].

The total graph T(G) of the graph G is a graph whose vertex set is the union of V(G) and E(G) such that ab∈ETG if and only if a and b are either adjacent or incident in G [8]. Let u, v, and w be the variables having values + or −. The transformation graph Guvw is a graph whose vertex set is the union of V(G) and E(G), and ab∈E(Guvw) if and only if

a,b∈V(G); then u=+ or u=- if a and b are adjacent or nonadjacent in G, respectively;

a,b∈E(G); then v=+ or v=- if a and b are adjacent or nonadjacent in G, respectively;

a∈V(G) and b∈E(G); then w=+ or w=- if a and b are incident or nonincident in G, respectively.

There are eight different transformations of the given graph G. For instance, G+++ is the total graph T(G) of G with number of vertices n+e and number of edges 1/2M1(G)+2e, and G--- is the complement of total graph G+++. For other transformations of graph, G++-, G+-+, and G+-- are the complements of G--+, G-+-, and G-++, respectively.

The concepts of semitotal-point graph and semitotal-line graph are introduced by Sampathkumar and Chikkodimath [9]. The semitotal-point graph T1(G) is a graph whose vertex set is the union of V(G) and E(G), and ab∈E(T1(G)) if and only if (i) a and b are adjacent vertices in G or (ii) one is a vertex of G and the other is an edge of G incident to it. Thus, semitotal-point graph has n+e number of vertices and 3e number of edges.

The semitotal-line graph T2(G) is a graph whose vertex set is the union of V(G) and E(G), and ab∈E(T2(G)) if and only if (i) a and b are adjacent edges in G and (ii) one is a vertex of G and the other is an edge of G incident to it. Thus, semitotal-line graph has n+e number of vertices and 1/2M1(G)+e 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 G-+-. Yi and Wu [14] determined the connectivity, the Hamiltonian, and the independence number of G++-.

In this paper, we obtain lower and upper bounds for the general sum-connectivity indices of the above-defined transformation graphs.

2. 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.

Theorem 1.

For α<0, we have γ1≤χα(T1(G))≤γ2, where (4)γ1=2αχαG+2α+1eΔG+1α,γ2=2αχαG+2α+1eδG+1α;the equalities hold if and only if G is a regular graph.

Proof.

Since T1(G) has n+e vertices and 3e edges, it holds that(5)χαT1G=∑ab∈ET1GdT1Ga+dT1Gbα=∑ab∈ET1G,a,b∈VGdT1Ga+dT1Gbα+∑ab∈ET1G,a∈VG,b∈EGdT1Ga+dT1Gbα.

Note that if a∈V(G) then dT1(G)(a)=2dG(a) and if a∈E(G) then dT1(G)(a)=2. It is clear that δ(G)≤dG(a) and Δ(G)≥dG(a). And these equalities hold if and only if G is a regular graph. Therefore,(6)χαT1G=2α∑ab∈ET1G,a,b∈VGdGa+dGbα+2α∑ab∈ET1G,a∈VG,b∈EGdGa+1α≥2αχαG+2α+1eΔG+1α.

Similarly, we can compute(7)χαT1G≤2αχαG+2α+1eδG+1α.The two equalities in (6) and (7) obviously hold if and only if G and H are regular, respectively.

Example 2.

By Theorem 1, the general sum-connectivity indices of some semitotal-point graphs are given below:

n8α+2×6α-3×8α≤χαT1Pn≤22αn2α+2+2×6α-3×8α-22α+1.

χαT1Cn=2αn4α+2×3α.

χαT1Kn=2αn(n-1)2α-1n-1α+nα.

Theorem 3.

If α<0 then γ1≤χαT2G≤γ2, where(8)γ1=22α-1M1GΔαG+eΔαG2×3α-4α,γ2=22α-1M1GδαG+eδαG2×3α-4α;the equalities hold if and only if G is a regular graph.

Proof.

Since VT2G=n+e and ET2G=1/2M1(G)+e, we have(9)χαT2G=∑ab∈ET2GdT2Ga+dT2Gbα=∑ab∈ET2G,a,b∈EGdT2Ga+dT2Gbα+∑ab∈ET2G,a∈VG,b∈EGdT2Ga+dT2Gbα.Note that if a∈E(G) then dT2(G)(a)=dG(wi)+dG(wj) and if a∈V(G) then dT2(G)(a)=dG(a). Therefore, we have(10)χαT2G=∑wiwj∈EG,wjwk∈EG,wi≠wkdGwi+dGwj+dGwj+dGwkα+∑ab∈ET2G,a∈VG,b=ax∈EG,x∈VGdGa+dGa+dGxα=∑wiwj∈EG,wjwk∈EG,wi≠wkdGwi+2dGwj+dGwkα+∑a∈VG,b=ax∈EG,x∈VGdGa+dGa+dGxα.Since dG(a)≥δ(G) and dG(a)≤Δ(G), each equality holds if and only if G is a regular graph.

After simplification we get(11)χαT2G≥4ΔGα∑wiwj∈EG,wjwk∈EG,wi≠wk1+3ΔGα∑ab∈ET2G,a∈VG,b=ax∈EG,x∈VG1=4ΔGα·ET2G-2e+3ΔGα·2e=22α-1M1GΔαG+eΔαG2×3α-4α.

Similarly, we can calculate(12)χαT2G≤22α-1M1GδαG+eδαG2×3α-4α.

Obviously the equalities in (11) and (12) hold if and only if G is a regular graph.

Example 4.

By Theorem 3, the general sum-connectivity indices of some semitotal-line graphs are given below:

2αn4α+2×3α-2α+14α-3α≤χαT2Pn≤n4α+2×3α-24α+2×3α.

χαT2Cn=2αn4α+2×3α.

χαT2Kn=nαn-1α+122α-1n+22α+3α.

Theorem 5.

Let α<0. Then γ1≤χαTG≤γ2, where(13)γ1=2αχαG+22α-1M1GΔαG+4αeΔαG,γ2=2αχαG+22α-1M1GδαG+4αeδαG;the equalities hold if and only if G is a regular graph.

Proof.

Since VTG=n+e and ETG=1/2M1(G)+2e, we have(14)χαTG=∑ab∈ETGdTGa+dTGbα=∑ab∈ETG,a,b∈VGdTGa+dTGbα+∑ab∈ETG,a,b∈EGdTGa+dTGbα+∑ab∈ETG,a∈VG,b∈EGdTGa+dTGbα.

Note that dT(G)(a)=2dG(a) for a∈V(G) and dT(G)(a)=dG(wi)+dG(wj) for a∈E(G). So(15)χαTG=2α∑ab∈EG,a,b∈VGdGa+dGbα+∑a=wiwj∈EG,b=wjwk∈EG,wi≠wkdGwi+dGwj+dGwj+dGwkα+∑b=ax∈EG,a∈VG,x∈VG2dGa+dGa+dGxα=2α·χαG+∑a=wiwj∈EG,b=wjwk∈EG,wi≠wkdGwi+2dGwj+dGwkα+∑b=ax∈EG,a∈VG,x∈VG2dGa+dGa+dGxα.Note that dG(a)≤Δ(G) and dG(a)≥δ(G). The equalities hold if and only if G is a regular graph.

After simplification, we get (16)χαTG≥2αχαG+4ΔGα12M1G-e+4ΔGα2e=2αχαG+22α-1ΔαGM1G+22αeΔαG.

Similarly, we can compute (17)χαTG≤2αχαG+22α-1δαGM1G+22αeδαG.

Since χα(G)≥2αeΔα(G), we can also write the results above as(18)4αδαG12M1G+2e≤χαTG≤4αΔαG12M1G+2e.

Thus, if G is a regular graph, then we obtain the equality in (16), (17), and (18).

Example 6.

By Theorem 5, the general sum-connectivity indices of some total graphs are given below:

Let α<0. Then γ1≤χα(G---)≤γ2, where(19)γ1=2α12M1G+2ee+n-1-2ΔαGα,γ2=2α12M1G+2ee+n-1-2δαGα;the equalities hold if and only if G is a regular graph.

Proof.

For a given graph G, since G---≅G+++¯ and G+++=T(G), then VG---=n+e, EG---=e+n2-1/2M1(G)-2e, and 2Δ(G¯)=e+n-1-2Δ(G). Using these values, we can compute the required results.

Theorem 8.

Let α<0. Then γ1≤χα(G++-)≤γ2, where(20)γ1=2αeα+1+2α12M1G-en-4+2ΔGα+e+n2-n2-e2-2e·e+n-4+2ΔGα,γ2=2αeα+1+2α12M1G-en-4+2δGα+e+n2-n2-e2-2e·e+n-4+2δGα;the equalities hold if and only if G is a regular graph.

Proof.

Since VG++-=n+e and EG++-=e+n2-n2-e2+1/2M1(G)-2m, (21)χαG++-=∑ab∈EG++-dG++-a+dG++-bα=∑ab∈EG++-,a,b∈VGdG++-a+dG++-bα+∑ab∈EG++-,a,b∈EGdG++-a+dG++-bα+∑ab∈EG++-,a∈VG,b∈EGdG++-a+dG++-bα.Note that if a∈V(G) then dG++-(a)=e and if a∈E(G) then dG++-(a)=dG(wi)+dG(wj)+n-4(22)χαG++-=∑ab∈EG,a,b∈VG2eα+∑a=wiwj∈EG,b=wjwk∈EG,wi≠wkdGwi+dGwj+n-4+dGwj+dGwk+n-4α+∑b=xy∈EG,a∈VG,a∉x,ye+dGx+dGy+n-4α=2αeα+1+∑wiwj∈EG,wjwk∈EG2n-8+dGwi+2dGwj+dGwkα+∑b=xy∈EG,a∈VG,a∉x,ye+n-4+dGx+dGyα.

Note that dG(a)≤Δ(G) and dG(a)≥δ(G). The equalities hold if and only if G is a regular graph. After simplification, we get (23)χαG++-≥2αeα+1+2α12M1G-en-4+2ΔGα+e+n2-n2-e2-2ee+n-4+2ΔGα.

Similarly, we can compute (24)χαG++-≤2αeα+1+2α12M1G-en-4+2δGα+e+n2-n2-e2-2ee+n-4+2δGα.

The equalities in (23) and (24) obviously hold if and only if G is a regular graphs.

Theorem 9.

Let α<0. Then γ1≤χα(G-+-)≤γ2, where(25)γ1=2αe+n-1-2ΔGαn2-e+2αn-4+2ΔGα12M1G-e+e+2n-5αe+n2-n2-e2-2e,γ2=2αe+n-1-2δGαn2-e+2αn-4+2δGα12M1G-e+e+2n-5αe+n2-n2-e2-2e;the equalities hold if and only if G is a regular graph.

Proof.

Since VG-+-=n+e and EG-+-=e+n2-e2+1/2M1(G)-4e,(26)χαG-+-=∑ab∈EG-+-dG-+-a+dG-+-bα=∑ab∈EG-+-,a,b∈VGdG-+-a+dG-+-bα+∑ab∈EG-+-,a,b∈EGdG-+-a+dG-+-bα+∑ab∈EG-+-,a∈VG,b∈EGdG-+-a+dG-+-bα.

Note that dG-+-(a)=e+n-1-2dG(a) for a∈V(G) and dG-+-(a)=dG(wi)+dG(wj)+n-4 for a∈E(G). Then(27)χαG-+-=∑ab∈EG-+-,a,b∈VGe+n-1-2dGa+e+n-1-2dGbα+∑wiwj∈EG,wjwk∈EG,wi≠wkdGwi+dGwj+n-4+dGwj+dGwk+n-4α+∑ab∈EG-+-,a∈VG,b=xy∈EGe+n-1-2dGa+dGx+dGy+n-4α=2α∑ab∉EG,a,b∈VGe+n-1-dGa-dGbα+∑wiwj∈EG,wjwk∈EG,wi≠wk2n-8+dGwi+2dGwj+dGwkα+∑b=xy∈EG,a∉x,y,a∈VGe+2n-5-2dGa+dGx+dGyα.Note that dG(a)≤Δ(G) and dG(a)≥δ(G). The equalities hold if and only if G is a regular graph.

After simplification, we get(28)χαG-+-≥2αe+n-1-2ΔGαn2-e+2αn-4+2ΔGα12M1G-e+e+2n-5αe+n2-n2-e2-2e.Similarly, we calculate(29)χαG-+-≤2αe+n-1-2δGαn-4+2δGαn2-e+2α12M1G-e+e+n2-n2-e+2n-5αe2-2e.If G is a regular graph then we obtain the equalities in (28) and (29).

In fully analogous manner, we also arrive at the following.

γ1≤χαG-++≤γ2, where (32)γ1=2αn2-en-1α+4αΔαG12M1G-e+2en-1+2ΔGα,γ2=2αn2-en-1α+4αδαG12M1G-e+2en-1+2δGα;

γ1≤χαG+-+≤γ2, where (33)γ1=22αeΔαG+e2-12M1G+e3-2ΔGα+2ee+3α,γ2=22αeδαG+e2-12M1G+e3-2δGα+2ee+3α.

In all the above cases, the equalities hold if and only if G is a regular graph, respectively.
3. Conclusion

In this paper, we obtain the sharp lower and upper bounds for general sum-connectivity indices of the semitotal-point graph, the semitotal-line graph, the total graph, and the eight distinct transformation graphs Guvw, where u,v,w∈+,- in terms of the order, minimum degree, and maximum degree of a graph. Moreover, the extremal graphs achieving these bounds have been described.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research is supported by the Fundamental Research Funds for the Central Universities (nos. 2652015193 and 2652017146) and NSFC of China (no. 11701530).

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