SHAFQAT Ali

(Department of Mathematics，Minhaj University，Lahore，Pakistan 54000)

Abstract: With the help of topological index, it′s helpful to understand the numerical quantities of concerned family of graphs. The aim of the present report is to determine the degree-based topological indices of the line graph of the subdivision of rooted product of cycles with paths. By means of edge dividing and graph structure analysis, we computed Harmonic index, Randic type indices, symmetric division index, atomic-bond-connectivity index, harmonic-index, symmetric division index, geometric-arithematic index, generalized reverse-Randic index, and inverse sum index of underlined family of graphs.

Key words: line graph；subdivision graph；topological index；Zagreb index

1 Background Knowledge

The concept of rooted product graph was introduced in 1978 by Godsil and McKay[1]. Given a graphGof ordern(G) and a graphHwith root vertexv, the rooted product graphG{H} is defined as the graph obtained fromGandHby taking one copy ofGandn(G) copies ofHand identifying thei-th vertex ofGwith the root vertexvin the ith copy ofHfor everyi∈{1, 2,…,n(G)}. IfHorGis a trivial graph, thenG{H} is equal toGorH, respectively. In this paper we aim to study the rooted product of cycles with paths.

The line graph of an undirected graphGis another graphL(G) that represents the adjacencies between edges ofG[2].

The real number attached with the graph of chemical structure is known ad topological index. The theory of topological indices begun in 1947, when Wiener index was introduced[3]. After this huge amount of topological indices are introduced to study graphs, see for example[4]. Here we study some of them. From now to on word, we considerGto be connected and simple graph. Now we give some definitions of topological indices that can be found in[5-6].

The Symmetric-division-index ofGis：

The Harmonic-index ofGis：

The Inverse-Sum index ofGis：

The generalized Randic index ofGis：

The generalized reverse-Randic index ofGis：

RRα(G)=∑(du×dv)α.

The atomic-bond-connectivity index ofGis：

The geometric-arithematic index ofGis：

The modified-Randic index ofGis：

In this paper, we computed all above defined degree-based topological indices for the line graph of the subdivision graph of rooted product of cycles with paths.

2 Main Results

In this section, we will present our main computational results.

2.1 Line Graph of the Subdivision Graph of Cn{Pk} for k>1

The line graph of rooted product of cycle and path fork>1 is presented in Figure 1.

Figure 1 Line Graph of the Subdivision Graph of Cn{Pk}

The edge partition forCn{Pk} is presented in Table 1.

Table 1 Edge Partition of E(Cn{Pk})

Theorem1ForCn{Pk} fork>1, the Harmonic index is：

ProofUsing the edge partition given in Table 1, we have following computation of Harmonic index：

Theorem2ForCn{Pk} fork>1, the inverse sum index is：

ProofUsing the edge partition given in Table 1, we have following computation of inverse-sum index：

Theorem3ForCn{Pk} fork>1, the generalized Randic index is：

ProofUsing the edge partition given in Table 1, we have following computation of generalized Randic index：

Theorem4ForCn{Pk} fork>1, the inverse generalized Randic index is：

RRα(Cn{Pk})=n(2α+(2k-3)4α+6α+4.9α).

ProofUsing the edge partition given in Table 1, we have following computation of the inverse generalized Randic index：

=2α|E1(Cn{Pk})|+4α|E2(Cn{Pk})|+6α|E3(Cn{Pk})|+9α|E4(Cn{Pk})|

=2α(n)+4α(2nk-3n)+6α(n)+9α(4n)

=n(2α+(2k-3)4α+6α+4.9α).

Theorem5ForCn{Pk} fork>1, the symmetric division index is：

ProofUsing the edge partition given in Table 1, we have following computation of symmetric division index：

Theorem6ForCn{Pk} fork>1, the Atomic bound connectivity index is：

ProofUsing the edge partition given in Table 1, we have following computation of symmetric division index：

Theorem7ForCn{Pk} fork>1, the geometric arithmetic index is：

ProofUsing the edge partition given in Table 1, we have following computation of Geometric Arithematic index：

Theorem8ForCn{Pk}， modified Randic index is：

2.2 Line Graph of the Subdivision Graph of Cn{Pk}for k=1

The line graph of the subdivision graph ofCn{Pk} fork=1 is shown in Figure 2.

Figure 2 Line Graph of the Subdivision Graph of Cn{Pk} for k=1

The edge partition forCn{Pk} is in Table 2.

Table 2 Edge partition for Cn{Pk} for k=1

The following results can be obtained immediately from the edge partition of the line graph of the subdivision graph ofCn{Pk} fork=1.

Theorem12ForCn{Pk} fork=1， the inverse generalized Randic index isRRα(Cn{Pk})=n(3α+4.9α).

3 Conclusions

In this paper, we computed several degree-based topological indices of line graph of subdivision graph of rooted product of cycles with paths. We computed our results with the help of edge partition of based on the degree of end vertices of edges. Our results can be helpful to understand the properties of concerned family of graphs. It is interesting to compute the distance based polynomials and indices for the family of graphs studied in this paper.