The study of line graphs of subdivision graphs of some rooted product graphs via K-Banhatti indices

1. Introduction

In this paper, we consider those graphs which are simple and connected. Let $G=(V,E)$ be a graph (molecular) with vertex set $V=V(G)$, and edge set $E=E(G)$. The cardinality of the set V and E is known as the order, and the size of the graph G, respectively. The number of edges incident to a vertex $x\in V$ is called the vertex degree, denoted by $d_x$. The degree of an edge e, denoted by $d_e$, is defined by $d_e=d_x+d_y-2$ where $e=xy$. Further, we write $x\sim e$. If $e=xy$ in an edge in a graph G, a new vertex z is inserted between the vertices x and y and the edge $xy$ is replaced by two new edges $xz$ and $zy$. This is known as the subdivision of an edge. Likewise, the subdivision graph $S(G)$ is formed from the subdivision of all the edges in G. The line graph, denoted by $L(G)$, of a graph G is the graph with $V(L(G))=E(G)$, and $e_1{e}_2\in E(L(G))$ if and only if $e_1$ and $e_2$ share a common endpoint in G. Further, $L(S(G))$ is the line graph of $S(G)$.

In the literature, a number of topological indices have been defined and studied. Kulli^4,5 proposed some novel degree-based topological indices namely, first K-Banhatti index, second K-Banhatti index, modified first K-Banhatti index, modified second K-Banhatti index, and harmonic K-Banhatti index.

The first K-Banhatti index of a graph G is defined as

the second K-Banhatti index of G is defined as the modified first K-Banhatti index of G is defined as the modified second K-Banhatti index of G is defined as the harmonic K-Banhatti index of G is defined as where $x\sim e$ means that the vertex x, and the edge e are incident in the graph G.

For more properties and different versions of those indices, readers can visit literature.^{4,6,7,8,9,10,11,12,13}

In particular, in Ref. 14, the exact values of the Shultz index of the subdivision graph of the wheel, tadpole, ladder, and helm graphs have been found. The Zagreb indices of the line graph with subdivision of these structures are studied in Ref. 15. In Ref. 16, the authors have found the values of the general sum-connectivity index and co-index of the line graph of these graphs with subdivision. In Ref. 17, $ABC4$ and $GA5$ indices of the line graph of the wheel, ladder, and tadpole graphs by means of the definition of subdivision have been determined. In Ref. 18, the $(a,b)$-Zagreb index of line graphs of subdivision graphs of some molecular structures have been studied.

In the following section, we will study the K-Banhatti indices of line graph of the subdivision graphs of the rooted graphs $C_n\{P_{k+1}\},C_n\{S_{m+1}\}$ and $C_{i,r}\{P_{k+1}\}$.

2. Examples

In this section, we will discuss the calculations of the K-Banhatti indices of $C_3\{P_2\}$ and the line graph of its subdivision (see Figs. 1 and 2).

Let G be the graph of $C_3\{P_2\}$. Since each of the vertices of G is of degree either one, two or three, the vertex set of G has the following three partitions with respect to degree:

and The edge set of G has three partitions based on the degree of the end vertices:

$$\begin{array}{l}{E}_{\{1,2\}}(G)=\{e=xy\in E(G){|}_{d_x=1,d_y=2}\},\\ E_{\{2,3\}}(G)=\{e=xy\in E(G){|}_{d_x=2,d_y=3}\},\\ E_{\{3,3\}}(G)=\{e=xy\in E(G){|}_{d_x=3,d_y=3}\}\end{array}$$

and the corresponding edge degrees are 1, 3, and 4, respectively. Further, the cardinality of each edge set will be Similarly, the edge partition for the line graph of the subdivision graph of $C_3\{P_2\}$ can be seen in Table 1.

Using Eqs. (1)–(5), we get the values of K-Banhatti indices of $C_3\{P_2\}$ and its line graph of the subdivision.

In the next section, we have found the general values of the K-Banhatti indices of $C_n\{P_{k+1}\}$ and the line graph of its subdivision.

3. Topological Indices of the Line Graph of the Subdivision Graph of Rooted Product of Some Graphs

Let $G_1,G_2,\dots ,G_n$ be the sequence of n rooted graph of G and H be a labeled graph with vertex set $V(H)=\{1,2,\dots ,n\}$. As defined in Ref. 19, the rooted product of H by G (that is, $H(G)=H(G_1,G_2,\dots ,G_n)$) is a graph obtained by identifying the root vertex of $G_i$ with the ith ($i=1,2,3,\dots ,n$) vertex of H.

3.1. K-Banhatti indices of line graph of the subdivision graph of $C_n\{P_{k+1}\}$

Here we consider the path ($P_n$), and cycle graph ($C_n$) with the order n. Let $C_n\{P_{k+1}\}$ be the cluster product of cycle and path of length n, and k, respectively. There are total $2n+2nk$ vertices among which the numbers of vertices of degree 1, 2, and 3 are n, $2nk-2n$, and $3n$, respectively. The graph of $C_n\{P_{k+1}\}$ and the line graph of the subdivision graph of $C_n\{P_{k+1}\}$ can be seen in Fig. 3 and their edge partitions are tabulated in Tables 2 and 3.

Theorem 1. Let G be a graph isomorphic to the line graph of subdivision graph of $C_n\{P_{k+1}\}$. Then,

$\mathrm{\text{(i)}}$	$$B_1(G)=\left\{\begin{array}{cc}64n\hfill & \mathrm{\text{if}}k=1,\hfill \\ 16n(3+k)\hfill & \mathrm{\text{if}}k>1,\hfill \end{array}\right.$$
$\mathrm{\text{(ii)}}$	$$B_2(G)=\left\{\begin{array}{cc}104n\hfill & \mathrm{\text{if}}k=1,\hfill \\ 90n+16nk\hfill & \mathrm{\text{if}}k>1,\hfill \end{array}\right.$$
$\mathrm{\text{(iii)}}$	$${}^m{B}_1(G)=\left\{\begin{array}{cc}\frac{176n}{105}\hfill & \mathrm{\text{if}}k=1,\hfill \\ \frac{59}{70}n+nk\hfill & \mathrm{\text{if}}k>1,\hfill \end{array}\right.$$
$\mathrm{\text{(iv)}}$	$${}^m{B}_2(G)=\left\{\begin{array}{cc}\frac{4n}{3}\hfill & \mathrm{\text{if}}k=1,\hfill \\ \frac{17}{18}n+nk\hfill & \mathrm{\text{if}}k>1,\hfill \end{array}\right.$$
$\mathrm{\text{(v)}}$	$$HB(G)=\left\{\begin{array}{cc}\frac{352}{105}\hfill & \mathrm{\text{if}}k=1,\hfill \\ \frac{161}{35}n+2nk\hfill & \mathrm{\text{if}}k>1.\hfill \end{array}\right.$$

Proof.

(i)	Case 1. $k=1$: $$B_1(G)=n(1+2+3+2)+4n(3+4+3+4)=8n+56n=64n.$$ Case 2. $k>1$: $$\begin{array}{l}{B}_1(G)=n(1+1+2+1)+(2nk-3n)\times (2+2+2+2)+n(2+3+3+3)\\ +n(3+4+3+4)\\ =16n(3+k).\end{array}$$
(ii)	Case 1. $k=1$: $$B_2(G)=n(2+6)+4n(12+12)=8n+96n=104n.$$ Case 2. $k>1$: $$B_2(G)=n(1+2)+(2nk-3n)(4+4)+n(6+9)+n(12+12)=90n+16nk.$$
(iii)	Case 1. $k=1$: $${}^m{B}_1(G)=n(1∕3+1∕5)+4n(1∕7+1∕7)=(8∕15)n+4n(2∕7)=(176∕105)n.$$ Case 2. $k>1$: $$\begin{array}{l}{}^m{B}_1(G)=n(1∕2+1∕3)+(2nk-3n)(1∕4+1∕4)+n(1∕5+1∕6)+n(1∕7+1∕7)\\ =n(59∕70)+nk.\end{array}$$
(iv)	Case 1. $k=1$: $${}^m{B}_2(G)=n(1∕2+1∕6)+4n(1∕12+1∕12)=n(2∕3)+(8∕24)n=(4∕3)n.$$ Case 2. $k>1$: $$\begin{array}{l}{}^m{B}_2(G)=n(1+1∕2)+(2nk-3n)(1∕4+1∕4)+n(1∕6+1∕9)+n(1∕12+1∕12)\\ =n(17∕18)+nk.\end{array}$$
(v)	Case 1. $k=1$: $$HB(G)=n(2∕3+2∕5)+4n(2∕7+2∕7)=(16∕15)n+n(16∕7)=(352∕105)n.$$ Case 2. $k>1$: $$\begin{array}{l}HB(G)=n(2∕2+2∕3)+(2nk-3n)(2∕4+2∕4)+n(2∕5+2∕6)+n(2∕7+2∕7)\\ =n(161∕35)+2nk.\end{array}$$

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3.2. K-Banhatti indices of line graph of the subdivision graph of $C_n\{S_{m+1}\}$

Let $S_{m+1}$ be the star graph. The cluster product of $C_n$ by $S_{m+1}$ is denoted by $C_n\{S_{m+1}\}$, and is obtained by identifying any pendant vertex of the ith copy of $S_{m+1}$ to the ith vertex of $C_n$ (see Fig. 4).

Theorem 2. Let G be a graph isomorphic to the line graph of subdivision graph of $C_n\{S_{m+1}\}$. Then,

$\mathrm{\text{(i)}}$	$$B_1(G)=mn(3m+5)+64n+mn(m+1)(3m+1),$$
$\mathrm{\text{(ii)}}$	$$B_2(G)=m^{2}n[(m+2)+2{(m+1)}^2]+n(m+2)(m+4)+96n,$$
$\mathrm{\text{(iii)}}$	$${}^m{B}_1(G)=\frac{3mn}{2(m+1)}+\frac{n(3m+8)}{(m+5)(2m+3)}+\frac{mn(m+1)}{(3m+1)}+\frac{8}{7}n,$$
$\mathrm{\text{(iv)}}$	$${}^m{B}_2(G)=\frac{n(8m+11)}{3(m+1)}+\frac{n(m+4)}{3(m+2)(m+1)},$$
$\mathrm{\text{(v)}}$	$$HB(G)=\frac{3mn}{(m+1)}+\frac{2n(3m+8)}{(m+5)(2m+3)}+\frac{2mn(m+1)}{(3m+1)}+\frac{16}{7}n.$$

Proof.

(i)	$$\begin{array}{l}{B}_1(G)=nm(1+m+2m+1)+n(3+2(m+2)+m+1)+4n(14)+\frac{mn(m+1)}{2}(6m+2)\\ =mn(3m+5)+64n+mn(m+1)(3m+1).\end{array}$$
(ii)	$$\begin{array}{l}{B}_2(G)=mn(m+m^2+m)+n(3m+6+(m+1)(m+2)+4n(24)+\frac{mn(m+1)}{2}\times (4m(m+1))\\ =m^{2}n[(m+2)+2{(m+1)}^2]+n(m+2)(m+4)+96n.\end{array}$$
(iii)	$$\begin{array}{l}{}^m{B}_1(G)=mn\left(\frac{1}{m+1}+\frac{1}{2m+1}\right)+4n\left(\frac{2}{7}\right)+n\left(\frac{1}{m+5}+\frac{1}{2m+3}\right)+\frac{mn(m+1)}{2}\frac{2}{3m+1}\\ =\frac{3mn}{2(m+1)}+\frac{n(3m+8)}{(m+5)(2m+3)}+\frac{mn(m+1)}{(3m+1)}+\frac{8}{7}n.\end{array}$$
(iv)	$${}^m{B}_2(G)=mn\left(\frac{1}{m}+\frac{1}{m(m+1)}\right)+4n\frac{1}{6}+\frac{mn(m+1)}{2}\frac{2}{2m(m+1)}=\frac{n(8m+11)}{3(m+1)}+\frac{n(m+4)}{3(m+2)(m+1)}.$$
(v)	$$\begin{array}{l}HB(G)=2mn\left(\frac{1}{m+1}+\frac{1}{2m+1}\right)+8n\left(\frac{2}{7}\right)+2n\left(\frac{1}{m+5}+\frac{1}{2m+3}\right)+2\frac{mn(m+1)}{3m+1}\\ =\frac{3mn}{(m+1)}+\frac{2n(3m+8)}{(m+5)(2m+3)}+\frac{2mn(m+1)}{(3m+1)}+\frac{16}{7}n.\end{array}$$

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3.3. K-Banhatti indices of line graph of the subdivision graph of $C_{i,r}\{P_{k+1}\}$

Let H be a labeled graph with order n and G be a sequence of k rooted graphs $G_1,G_2,\dots ,G_k$. Then the ith vertex rooted product of H by G, denoted by $H_i\{G_1,G_2,\dots ,G_k\}$, is generated by identifying the root vertex of every $G_i$ to the vertex of H for all $i\in 1,2,\dots ,k.$ In the special case when the components $G_1,G_2,\dots ,G_k$ are mutually isomorphic to a graph L, the ith vertex rooted product of H by G is denoted by $H_{i,k}\{L\}$ and is called the ith vertex cluster product of H by L.²⁰

The graph of $C_{i,r}\{P_{k+1}\}$ can be seen in Fig. 5 and its edge partition is tabulated in Tables 5 and 6.

Theorem 3. Let G be a graph isomorphic to the line graph of subdivision graph of $C_{i,r}\{P_{k+1}\}$. Then,

$\mathrm{\text{(i)}}$	$$B_1(G)=\left\{\begin{array}{cccc}r(3r+5)+8(2n-3)\hfill & +2(3r+8)\hfill & +(r^2+3r+2)(3r+4)\hfill & \mathrm{\text{if}}k=1,\hfill \\ 5r+8(2n+2kr-3r-3)\hfill & +(r+4)(3r+8)\hfill & +(r^2+3r+2)(3r+4)\hfill & \mathrm{\text{if}}k>1,\hfill \end{array}\right.$$
$\mathrm{\text{(ii)}}$	$$B_2(G)=\left\{\begin{array}{cccc}r(r+1)(r+3)+8(2n-3)\hfill & +2(r+2)(r+4)\hfill & +2{(r+1)}^2{(r+2)}^2\hfill & \mathrm{\text{if}}k=1,\hfill \\ 3r+8(2n+2kr-3r-3)\hfill & +{(r+2)}^2(r+4)\hfill & +2{(r+1)}^2{(r+2)}^2\hfill & \mathrm{\text{if}}k>1,\hfill \end{array}\right.$$
$\mathrm{\text{(iii)}}$	$${}^m{B}_1(G)=\left\{\begin{array}{ccc}\frac{r(3r+5)}{(r+2)(2r+3)}+\frac{(2n-3)}{2}\hfill & +\frac{2}{r+2}+\frac{r^2+3r+2}{3r+4}\hfill & \mathrm{\text{if}}k=1,\hfill \\ \frac{5}{6}r+\frac{1}{2}(2n+2kr-3r-3)\hfill & +\frac{3r+8}{2(r+4)}+\frac{r^2+3r+2}{3r+4}\hfill & \mathrm{\text{if}}k>1,\hfill \end{array}\right.$$
$\mathrm{\text{(iv)}}$	$${}^m{B}_2(G)=\left\{\begin{array}{ccc}\frac{r(r+3)}{(r+1)(r+2)}+\frac{1}{2}(4n-5)\hfill & +\frac{2(r+4)}{{(r+2)}^2}+\frac{1}{2}(2n-3)\hfill & \mathrm{\text{if}}k=1,\hfill \\ \frac{1}{2}(3r+1)\hfill & +\frac{1}{2}(2n+2kr-3r-3)\hfill & +\frac{r+4}{2(r+2)}\hfill & \mathrm{\text{if}}k>1,\hfill \end{array}\right.$$
$\mathrm{\text{(v)}}$	$$HB(G)=\left\{\begin{array}{ccc}\frac{2r(3r+5)}{(r+2)(2r+3)}+(2n-3)\hfill & +\frac{4}{r+2}+\frac{2(r^2+3r+2)}{3r+4}\hfill & \mathrm{\text{if}}k=1,\hfill \\ \frac{5}{3}r+(2n+2kr-3r-3)\hfill & +\frac{3r+8}{(r+4)}+\frac{2(r^2+3r+2)}{3r+4}\hfill & \mathrm{\text{if}}k>1.\hfill \end{array}\right.$$

Proof.

(i)	Case 1. $k=1$: $$\begin{array}{l}{B}_1(G)=r(r+3+2r+2)+8(2n-3)+2(3r+8)+\frac{r^2+3r+2}{2}(3r+4+3r+4)\\ =r(3r+5)+8(2n-3)+2(3r+8)+(r^2+3r+2)(3r+4).\end{array}$$ Case 2. $k>1$: $$\begin{array}{l}{B}_1(G)=r(5)+8(2n+2kr-3r-3)+(r+2)[3r+8]\\ =\frac{1}{2}(r^2+3r+2)[3r+4+3r+4]\\ =5r+8(2n+2kr-3r-3)+(r+4)(3r+8)+(r^2+3r+2)(3r+4).\end{array}$$
(ii)	Case 1. $k=1$: $$\begin{array}{l}{B}_2(G)=r[(r+1)+(r+1)(r+2)]+8(2n-3)+2[2(r+2)+{(r+1)}^2]+\frac{r^2+3r+2}{2}[4(r+1)(r+2)]\\ =r(r+1)(r+3)+8(2n-3)+2(r+2)(r+4)+2{(r+1)}^2{(r+2)}^2.\end{array}$$ Case 2. $k>1$: $$\begin{array}{l}{B}_2(G)=3r+8(2n+2kr-3r-3)+(r+2)[2(r+2)+{(r+2)}^2]+\frac{r^2+3r+2}{2}[4(r+1)(r+2)]\\ =3r+8(2n+2kr-3r-3)+{(r+2)}^2(r+4)+2{(r+1)}^2{(r+2)}^2.\end{array}$$
(iii)	Case 1. $k=1$: $$\begin{array}{l}{}^m{B}_1(G)=r\left(\frac{1}{r+2}+\frac{1}{2r+3}\right)+\frac{2n-3}{2}+2\left(\frac{1}{2(r+2)}+\frac{1}{2(r+2)}\right)+\frac{r^2+3r+2}{2}\left(\frac{2}{3r+4}\right)\\ =\frac{r(3r+5)}{(r+2)(2r+3)}+\frac{(2n-3)}{2}+\frac{2}{r+2}+\frac{r^2+3r+2}{3r+4}.\end{array}$$ Case 2. $k>1$: $$\begin{array}{l}{}^m{B}_1(G)=r\left(\frac{1}{2}+\frac{1}{3}\right)+\frac{2n+2kr-3r-3}{2}+(r+2)\left(\frac{1}{(r+4)}+\frac{1}{(2r+4)}\right)+\frac{r^2+3r+2}{2}\left(\frac{2}{3r+4}\right)\\ =\frac{5}{6}r+\frac{1}{2}(2n+2kr-3r-3)+\frac{3r+8}{2(r+4)}+\frac{r^2+3r+2}{3r+4}.\end{array}$$
(iv)	Case 1. $k=1$: $$\begin{array}{l}{}^m{B}_2(G)=r\left(\frac{1}{1+r}+\frac{1}{(r+1)(r+2)}\right)+\frac{2n-3}{2}+2\left(\frac{1}{2(r+2)}+\frac{1}{{(r+1)}^2}\right)+\frac{r^2+3r+2}{2}\frac{2}{(r+1)(r+2)}\\ =\frac{r(r+3)}{(r+1)(r+2)}+\frac{1}{2}(4n-5)+\frac{2(r+4)}{{(r+2)}^2}+\frac{1}{2}(2n-3).\end{array}$$ Case 2. $k>1$: $$\begin{array}{l}{}^m{B}_2(G)=r\left(1+\frac{1}{2}\right)+\frac{2n+2kr-3r-3}{2}+(r+2)\left(\frac{1}{2(r+2)}+\frac{1}{{(r+2)}^2}\right)+\frac{(r+1)(r+2)}{2(r+1)(r+2)}\\ =\frac{1}{2}(3r+1)+\frac{1}{2}(2n+2kr-3r-3)+\frac{r+4}{2(r+2)}.\end{array}$$
(v)	Case 1. $k=1$: $$HB(G)=\frac{2r(3r+5)}{(r+2)(2r+3)}+(2n-3)+\frac{4}{r+2}+\frac{2(r^2+3r+2)}{3r+4}.$$ Case 2. $k>1$: $$HB(G)=\frac{5}{3}r+(2n+2kr-3r-3)+\frac{3r+8}{(r+4)}+\frac{2(r^2+3r+2)}{3r+4}.$$

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4. Conclusion

In this paper, we compute the exact values of K-Banhatti indices namely, first K-Banhatti index, second K-Banhatti index, modified first K-Banhatti index, modified second K-Banhatti index, and harmonic K-Banhatti index of line graphs of subdivision graphs of some rooted product graphs (In particular, $C_n\{P_{k+1}\}$, $C_n\{S_{m+1}\}$, and $C_{i,r}\{P_{k+1}\}$). In the future, we would want to compute the exact values of K-Banhatti indices for some more graph operations.

Competing Interests

The authors declare that they have no competing interests.