阿比德·薩利姆，阿妮拉·哈尼夫，阿德南·阿斯拉姆

(1．巴哈丁扎卡利亞大學(xué) 純數(shù)學(xué)與應(yīng)用數(shù)學(xué)高級(jí)研究中心，木爾坦 60800；2．明哈吉大學(xué) 數(shù)學(xué)系，拉合爾 54000；3．拉合爾工程與科技大學(xué) 拉查納學(xué)院，拉合爾 54000)

0 Introduction

LetGbe a graph having the vertex setV(G) and the edge setE(G). The graphGis called connected set if there exist a connection between all pair of vertices of it. The degree of a vertexuis the number of vertices adjust to it and will be represented bydu. Throughout this paper,Gwill represent a connected graph,Vits vertex set,Eits edge set, anddvthe degree of its vertexv.

In mathematical chemistry, mathematical tools are used to solve problems arising in chemistry. Chemical graph theory is an important area of research in mathematically chemistry which deals with topology of molecular structure such as the mathematical study of isomerism and the development of topological descriptors or indices. TIs are real numbers attached with graph networks and graph of chemical compounds and has applications in quantitative structure-property relationships. TIs remain invariant upto graph isomorphism and help to predict many properties of chemical compounds, networks and nanomaterials, for example, viscosity, boiling points, radius of gyrations, etc without going to lab[1-4].

Other emerging field is Cheminformatics, which is helpful in guessing biological activity and chemical properties of nanomaterial and networks. In these investigations, some Physico-chemical properties and TIs are utilized to guess the behavior of chemical networks[5-9]. The definitions of known topological indices can be found in [10—12] and references therein.

The line graphL(G) of a simple graphGis obtained by associating a vertex with each edge of the graph and connecting two vertices with an edge iff the corresponding edges ofGhave a vertex in common. Properties of a graphGthat depend only on adjacency between edges may be translated into equivalent properties inL(G) that depend on adjacency between vertices.

In this paper we study the line graph of the Ladder graphs. We computed several degree-based topological indices of understudy families of graphs.

The ladder graph is a planar undirected graph with 2nvertices and 3n-2 edges. The ladder graph can be obtained as the Cartesian product of two path graphs, one of which has only one edge. In this section, letGbe the line graph of Ladder Graph. The line graph of ladder graph is given in Figure 1.

1 Methodology

There are three kinds of TIs:

1.Degree-based TIs；

2.Distance-based TIs；

3.Spectral-based TIs．

In this paper, we focus on degree-based TIs. To compute degree-based TIs of line graph of the Ladder graphs, firstly we drawn line graphs and then we divide the edge set of this line graphs into classes based on the degree of the end vertices and compute there cardinality. From this edge partition, we compute our desired results.

2 Main Results

In this section we gave our main results.

Theorem1LetGbe the line graph of Ladder graph, then we have:

ProofCase1n=2.

We can divide the edge set of the line graph of ladder graph into following three classes depending on each edge at the end vertices of the degree：

E1(G)={e=uv∈E(G);du=2 anddv=3};

E2(G)={e=uv∈E(G);du=3 anddv=3};

E3(G)={e=uv∈E(G);du=3 anddv=4}.

Now we have |E1(G)|=4, |E2(G)|=2, and |E3(G)|=4.

Case2n>2.

We can divide the edge set of the line graph of ladder graph into following three classes depending on the degree of end vertices of each edge:

E1(G)={e=uv∈E(G);du=2,dv=3};

E2(G)={e=uv∈E(G);du=3,dv=4};

E3(G)={e=uv∈E(G);du=dv=4}.

Now |E1(G)|=4,|E2(G)|=8, and |E3(G)|=6n-14.

Now by applying definitions and with the help of above edge division, we can compute our desired results.

Following results can also be proved in similar fashion.

Theorem2LetGbe the line graph ladder graph, then we have：

Theorem3LetGbe the line graph of ladder Graph, then we have：

Theorem4LetGbe a line graph of ladder graph, then we have：

ProofForm the information given in theorem 1, we have：

The desired results can be obtained easily with the help of Table 1 and Table 2.

Table 1 Edge Partition of Line Graph

Table 2 Edge Partition of Line Graph

Theorem5LetGbe the line graph of ladder graph, then we have:

Theorem6LetGis the wheel graph of ladder graph, then we have:

3 Conclusion

In present report, we computed several degree based topological indices of line graph of ladder graph. During the last two decades a large number of numerical graph invariants (topological indices) have been defined and used for correlation analysis in theoretical chemistry, pharmacology, toxicology, and environmental chemistry. Topological indices are used to guess properties of chemical compounds without going to wet lab. Almost all properties of a chemical compound can be obtained from the topological indices. In this way, our results are important for chemists and drug designers.