The -polynomial was introduced by Deutsch and Klavžar in 2015 as a graph polynomial to provide an easy way to find closed formulas of degree-based topological indices, which are used to predict physical, chemical, and pharmacological properties of organic molecules. In this paper we give general closed forms of the -polynomial of the generalized Möbius ladder and its line graph. We also compute Zagreb Indices, generalized Randić indices, and symmetric division index of these graphs via the -polynomial.
The -Polynomial and Topological Indices of Generalized Möbius Ladder and Its Line Graph]
The -Polynomial and Topological Indices of Generalized Möbius Ladder and Its Line Graph
Subject Classification (2010). 05C07, 92E10, 05C31
Keywords. -polynomial, Generalized Möbius ladder, Line graph of generalized Möbius ladder, Zagreb indices, Generalized Randić indices
The -polynomial was introduced by E. Deutsch and S. Klavžar in 2015 in  as a graph invariant to play a role for degree-based invariants parallel to the role the Hosoya polynomial plays for distance-based invariants.
The -polynomial has applications in mathematical chemistry and pharmacology. The most interesting application of the M-polynomial is that almost all degree-based graph invariants, which are used to predict physical, chemical, and pharmacological properties of organic molecules, can be recovered from it; for more information please see [5, 9, 13, 15, 28, 31, 32].
The -polynomial and related topological indices have been studied for several classes of graphs. In 2015 Deutsch and Klavžar gave M-polynomial, first Zagreb, and second Zagreb indices of polyomino chains, starlike trees, and triangulenes . In 2017 Mobeen et al gave -polynomial and several degree-based topological indices of titania nanotubes , of triangular Boron Nanotubes , and of jahangir graph . In 2017 Ajmal et al gave M-polynomial and topological indices of generalized prism network .
Several degree-based topological indices, which play important role in mathematical chemistry, can be recovered from the -polynomial: The most famous degree-based index is the Randić index and was introduced by Milan Randić in 1975 . It is often used in cheminformatics for investigations of organic compounds; for more information, please see [11, 16, 26]. Later in 1998, working independently, Amic et al  and Bollobas-Erdos  proposed the generalized Randić index; for more information, please see [17, 20]. Gutman and Trinajstić introduced first Zagreb and second Zagreb indices in 1972 . The augmented Zagreb index was proposed by Furtula et al. in 2010 in  and is useful for computing heat of information of alkanes [11, 19]. To know more about topological indices, their computing, and their applications we refer the reader to [4, 6, 7, 10, 23, 24, 25, 29, 31, 33].
This article is devoted to study the -polynomial. We not only give the general forms of the -polynomials of the generalized Möbius ladder and its line graph but also recover first Zagreb, second Zagreb, second modified Zagreb, generalized Randić, reciprocal generalized Randić, and symmetric division indices from them.
This section covers the definitions of graph, degree of a vertex, line graph, molecular graph, -polynomial, topological index, Zagreb indices, generalized Randić indices, and generalized Möbius ladder.
A graph is a pair , where is the set of vertices and the set of edges. The edge between two vertices and is denoted by . The degree of a vertex , denoted by is the number of edges incident to it. A path from a vertex to a vertex is a sequence of vertices and edges that starts from and stops at . The number of edges in a path is the length of that path. A graph is said to be connected if there is a path between any two of its vertices.
Figure 1: A connected graph with to
The line graph of a graph , written , is the graph whose vertices are edges of , and when and are adjacent edges of then is an edge of .
Figure 2: The line graph of the above graph
A molecular graph is a representation of a chemical compound in terms of graph theory. Specifically, molecular graph is a graph whose vertices correspond to (carbon) atoms of the compound and whose edges correspond to chemical bonds. For instance, Figure 1 represents the molecular graph of 1-bromopropyne ().
In the following by we shall mean a connected graph, its edge set, its vertex set, and its edge joining the vertices and .
 The M-polynomial of is
where is the number of edges of with and .
A function which assigns to every connected graph a unique number is called a graph invariant. Instead of the function it is custom to say the number as the invariant. An invariant of a molecular graph which can be used to determine structure-property or structure-activity correlation is called the topological index. A topological index is said to be degree-based if it depends on degrees of the vertices of the graph.
The following are definitions of some those degree-based indices that have connection with the -polynomial.
The first Zagreb, second Zagreb, and second modified Zagreb indices of are respectively
, , and .
The generalized Randić and reciprocal generalized Randić indices of are respectively
The symmetric division index of is
A remarkable property of the -polynomial is that all the above degree-based indices can be recovered from it, using the relations given in the following table.
|Index||Relation with the -polynomial|
|Modified Second Zagreb|
|Reciprocal Generalized Randić|
 Consider the Cartesian product of paths and with vertices and , respectively. Take a twist and identify the vertices with the vertices , , , respectively, and identify the edge , with the edge , , where . What we receive is the generalized Möbius ladder .
You can see in the following figure.
Figure 3: The grid form of the generalized Möbius ladder
The original form of is:
Figure 4: The generalized Möbius ladder
In this section the general closed formulas of the -polynomial of the generalized Möbius ladder and its line graph are given.
The -polynomial of the generalized Möbius ladder , is
Depending on degrees of the vertices, the edges of can be divided into three disjoint sets: , , and .
In order to count the number of elements in each of these sets we must consider the grid shape of . The degree-three vertices lie on the top and bottom rows of . Since in each such row there are vertices, the number of edges whose adjacent vertices are of degree 3 in each such row is . Thus, . For better understanding, let us have a look at, for instance, the grid shape of :
Figure 5: Grid of
The degree-three vertices on the top that are adjacent to degree-four vertices determine the edges for , and the degree-three vertices on the bottom that are adjacent to degree-four vertices determine the edges for ; observe that such edges appear vertically. It follows that such edges lie on the top and such edges lie on the bottom of the grid of . Hence .
The edges whose adjacent vertices have degrees 4 can be split into two types, horizontal and vertical. The horizontal edges determined by degree-four vertices are , where for each value of in takes values in . Hence, the number of horizontal edges is . The vertical edges determined by degree-four vertices are , where for each value of in takes values in . Hence, the number of vertical edges is . So, , and we are done. ∎
Let be the generalized Möbius ladder for . Then
Since the line graph of the generalized Möbius ladder have only vertices of degrees 4, 5, and 6, the edge set of can be divided into five disjoint sets: , , , , and .
In order to to count the number of elements in each we need to consider in terms of the grid shape of . The edges of degree 4 appear only on the top and bottom rows of . On the top row each degree-four vertex of lies on the edge of , and on the bottom row each degree-four vertex of lies on the edge of . Since the number of edges determined by degree-four vertices on the top row of is and the number of edges determined by degree-four vertices on the bottom row of is , . For better understanding, let us have a look at, for instance, the grid of along with its line graph:
Figure 6: Grid of
Figure 7: Line graph of
Each vertex of degree 4, which is connected to a vertex of degree 5, lies either on the top row or on the bottom row of of ; see for clarity the line graph of . Each such vertex is connected to two vertices of degree 5; each degree-five vertex of is determined by the edge , on the top and by the edge , on the bottom. It follows that the number of edges whose adjacent vertices have degrees 4 and 5 on the top of is . Hence, the total number of such edges is , i.e., .
There is no edge with adjacent vertices of degree 5; to convince yourself, you may see .
The degree-five vertices of that are connected to degree-six vertices of lie at the edges and for each column , of . Each of degree-five vertex of that lies at the edge of is connected with three vertices of degree 6, two are lying at the edges and and one is lying at the edge of . Since there are such vertices, the number of edges, whose adjacent vertices are of degree 5 and 6, determined by them is . Similarly, the number of edges, whose adjacent vertices are of degree 5 and 6, determined by degree-five vertices of that lie at the edges of is . Hence .
The edges of whose adjacent vertices have degrees 6 can be divided into three types: the edges that appear horizontally, the edges that appear vertically, and the edges that appear diagonally. Each horizontal edge of with adjacent vertices of degrees 6 appears in row of the grid of . Since in each row there are such edges and there are rows, the total number of horizontal edges in is . The degree-six vertices of that generate vertical edges in are determined by the vertices , of . Since each column of the grid of contains vertices, there are vertical edges in each column. Moreover, since there are columns, the total number of vertical edges in is . The degree-six vertices of that are connected diagonally to degree-six vertices of lie at the edges for each column , of . Since in each column there are such vertices and each vertex is diagonally connected to 4 vertices of degree 6, the number of edges, whose adjacent vertices are of degree 6, in each column is . Moreover, since there are columns, the total number of such edges is . It now follows that . This completes the proof. ∎
In this section we recover the first Zagreb, second Zagreb, modified second Zagreb, generalized Randić, reciprocal generalized Randić, and symmetric division indices from the -polynomials of the generalized Möbius ladder and its line graph.
The degree-based topological indices of are:
From the -polynomial of we get
The degree-based topological indices of are:
The proof is similar to the proof of Proposition \@setrefprop4.1. ∎
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Abdus Salam School of Mathematical Sciences, GC University, Lahore-Pakistan
School of Automation, Beijing Institute of Technology, Beijing-China
Abdus Salam School of Mathematical Sciences, GC University, Lahore-Pakistan