Some Bounds on the Modified Randic Index

In this paper, we present some new lower and upper bounds for the modified Randic index in terms of maximum, minimum degree, girth, algebraic connectivity, diameter and average distance. Also we obtained relations between this index with Harmonic and Atom-bond connectivity indices. Finally, as an application we computed this index for some classes of nano-structures and linear chains.


INTRODUCTION
In this paper is a simple connected graph, where is the set vertex of , and is the edge set of .There are many different kinds of chemical indices that some of them are distance based like Wiener index, some of them are based on degree like Randic index.This fact is emphasized in the recent survey [12] which contains uniform approach to the degree-based indices.
The Randic index was proposed by Milan Randic in 1975.This topological index was named Branching index, later called Randic index, which defined as where denote the degree of vertex .This index has been defined to measure the extent of branching of the carbon-atom skeleton of saturated hydrocarbons.Although Milan Randic showed that there is a good correlation between this index and physicochemical properties of alkanes such as boiling points, surface areas and energy levels [1,3,27].There are many applications in organic chemistry, medicinal chemistry and pharmacology that this index became one of the most interesting topic in graph theory which 4 books are devoted [10,[18][19]23].In 2011, Z.Dvorak proposed a modified of Randic index, defined as , that is more tractable from computational point of view.It is much easier to follow during graph modifications than Randic index see [5] for more details.In [2], the authors showed that for every graph with n vertices, R´(G) is at least 1 no more than and these bounds attained by stars and regular graph.Although they determined graphs with minimal and maximal value of R´(G) among all trees and unicylic graphs, in [4], the authors showed that for all connected graph the inequality holds where is the minimum eccentricity among all vertices of and the eccentricity of the vertex is the maximum distance from to any vertex.The maximum and minimim degree of a vertex in denoted by ∆(G) and δ(G), respectively.
The Laplacian matrix of G is defined as , where is the diagonal matrix of its vertex degree and is the adjacency matrix.Among all eigenvalues of the Laplacian matrix of G, one of the most popular is the second smallest, which was called the algebraic connectivity of a graph by fiedler [9 ] in 1973, and denoted by .In [22], the authors get relation between Randic index and algebraic connectivity.The girth of a graph G, denoted by is the minimum length of its cycles.In [21] the authors computed upper bound of Randic index with girth .Let be the average distance of G that defined as such that is the Wiener index defined as the sum of the lengths of the shortest path between all pairs of vertices and diameter of is the maximum distance over all pairs of vertices and of denoted by .In [30], the authors obtained relation between Randic index and diameter of a graph.
The edge cut of is a group of edges whose total removal renders the graph disconnected.The edge connectivity is the size of a smallest edge cut.In this paper, we obtain a new bounds for the modified Randic index in terms of girth, diameter and algebraic connectivity.In continue, we establish some relation between this index and harmonic index and ABC index.The harmonic index of graph G is defined as .The atom-bond connectivity index of a nontrivial graph G, denoted by , is defined as .For more information about harmonic and ABC index we refer the reader to see [7,11,26,29].

MAIN RESULTS
The aim of this section is to determine some new bounds for in terms of girth, diameter and algebraic connectivity minimum and maximum degree.Proof: Let be an edge in .Since is triangle-free, we have .Therefore regarding the definition of , we have .Furthermore, If is the vertex star , then = .
Conversely, we assume that but is not isomorphic to , then there must existan edge such that , implying that , a contradiction.This completes the proof.The equality holds if G .Proof: In [5], the authors proved that . ( It is easy to see that contains at least one cycle since minimum degree δ is at least 2, so the girth of is at least 3, this implies the inequality.Therefore the equalities hold if G , so the proof is now completed.Lemma 2.5: Let be a graph on vertices with the algebraic connectivity we have: , where λ denotes the edge connectivity of .Proof: see [9] for more details.
Theorem 2.6: Let be a graph on vertices and edge connectivity λ such that , we get the following inequality: and if then we have: ).
Proof: Due to the above Lemma, if the edge connectivity we have ) and by using inequality (1) we can obtain: To prove the second part, it is enough to apply lemma 2.5.
Theorem 2.7: Fix a positive integer n.Among all trees on vertices and maximum degree ∆, the maximum value of algebraic connectivity equals to .Proof: See [28 ].
Theorem 2.8: Among all trees, the maximum value of modified Randic index equals to .Proof: See [2].Theorem 2.9: Let T be a tree with vertices and algebraic connectivity , the following ineqaulity holds: ).
Proof: Due to Theorem 2.7and 2.8, we have: , , so we have:   The equality holds in the above inequality if and only if .

Computation of the modified Randic index of TUZC 6 (p,q) nanotubes:
A carbon nanotube is forming from a graphite sheet that is rolled up so that it has a zigzag edge.In this paper, we computed the modified Randic index for some families of polyhex nanotubes, armchair, Phenylenic Nanotorus, Polycyclic Aromatic Hydrocarbons and polyomino chain (Figs.2-8).
By considering the lattice of TUZC 6 [p,q], we denote the number of hexagon in the first row by p and the number of rows by q.In each row, there are 2p vertices and hence the number of vertices in this nanotube equals to 2pq.In [13] the authors obtained the hyper Wiener and Schultz indices of TUZC 6 [p,q] nanotube, in [14] the authors computed GA index for this nanotube and in [8], the computed some connectivity index and Zagreb index of nanotube.Now in this section, we compute the modified index of TUZC 6 [p,q].Fig. 2. The 2-Dimensional Lattice of TUZC 6 [7,6]

Set
, since in the graph of nanotube TUZC 6 (p,q), all of edges are in or , we need to obtain the number of and .

Lemma 3.1:
The number of equals to 4p and the number of equals to 3pq-5p.Proof: Consider the TUZC 6 [p,q] nanotube.At the first and last rows, there exist edges that every edge in these rows belong to , hence the number of equals to 4p.At the other rows there exist p edges that belong to and the number of these edges are q-1, 2p edges that every edges belong to and the number of these edges are q-2, hence the number of equals to 3pq-5p.Theorem 3.3: The modified Randic index of TUAC 6 [p,q] equals to .Proof: Like the previous theorem we have: = + = The next goal of this section is a computing a closed formula of the modified Randic index of TUC 4 C 8 [p,q] nanotube.In the structure of this nanotube there are pq horizontal regular squaroctagone lattice with 8pq+2p vertices and 12pq+p edges (Fig. 4).For more results about this nanotube see [15][16][17]24].In continue, we obtain R'(G) of a physico chemical structure of Phenylenic Nanotorus.This nano structure is V-Phenylenic Nanotorus VPHY[p,q].The structure of this nanotorus in terms of several C 4 C 6 C 8 net that composed of four and six membered rings such that every square is adjacent to two hexagones (Fig. 5).Theorem 3.5: For , the modified Randic index of V-Phenylenic Nanotori VPHY[p,q] equals to .Proof: Due to the general Figure of V-Phenylenic Nanotori VPHY[p,q], this nanotori has 6pq vertices, 9pq edges and all edges belong to f 3,3 , this implies that .At the next goal, we calculate the modified Randic index of hydrocarbon structures Polycyclic Aromatic Hydrocarbons (PAH n ).PAH n are a complex group of chemicals containing two or more aromatic rings.PAH n s are created when products like coal, oil, gas and garbage are burned but the burning process is not complete.The first member is Benzene (PAH 1 ) with six carbon and six hydrogen atoms and the second member is coronene (PAH 2 ) with 24 carbon and 12 hydrogen atoms (Fig. 6).By the Figure of the polycyclic aromatic hydrocarbon, it is easy to see that the general representation of has PAH n 6n 2 carbon and 6n hydrogen atoms.The modified Randic index of polyomino chain: A polyomino system is a finite 2-connected plane graph such that each interior face is surrounded by a regular square of length one.A polyomino chain is a polyomino system, in which the joining of the centers of its adjacent regular forms a path where is the center of the i-th square.Let be the set of polyomino chains with n squares, the subgraph of that induced by the vertices with degree 3 and n-2 squares, called a linear chain and denoted by (Fig. 7).The subgraph of induced by the vertices with degree bigger than 2 be a path with n-1 edges, called a zig-zag chain and denoted by (Fig. 8).In [31] the authors obtained Randic index of this graph.
A kink of a polyomino chain is any branched or angularly connected squares.A segment S of a polyomino chain is a maximal linear chain in the polyomino chains that include the kinks at its end.The number of squares in a segment denoted by A polyomino chains consist a sequence of segments ,…, , , with where +… = and n denote the number of squares of polyomino chain.In following, we assume that such that .

CONCLUSIONS
In this paper we achieved the lower and upper bounds for the modified Randic index in terms of girth, diameter and algebraic connectivity.Then we obtained a relation between this index with Harmonic and ABC indices.At the end of this paper we computed this index for some families of polyhex nanotubes TUZC 6 [p,q], TUAC 6 [p,q], TUC 4 C 8 [p,q], VPHY[p,q] nanotorus, Polycyclic Aromatic Hydrocarbons and polyomino chains for the first time.

Theorem 2 . 1 :
Let be a connected triangle-free graph with vertices and edges.Then we have: with equality if and only if is an (n+1)-vetrex star .

) Lemma 2 . 10 :. 2 . 12 :
Let be a connected graph with vertices and minimum degree δ Then it follows that .Proof: see[6] for more details.Theorem 2.11: Let G be a connected gaph with verices and minimum degree Then we get the following inequality: Proof: By the Inequality (1) and the above Lemma we have: Lemma If is a graph with vertices and minimum degree , then we have: Proof: See[20] for more details.

Theorem 2 . 13 :
Let be a connected graph with vertices and minimum degree then it follows that Proof: By inequality (1) and the above Lemma we have The proof is now complete.Now, we obtain relations between the modified Randic index, Harmonic and ABC indices.