ON THE GUTMAN INDEX OF THORN GRAPHS

In this paper, the relation between the Gutman inde x of a simple connected graph and its thorn graph is stablished and several special cases of the resu lt are examined. Results are applied to compute the Gutman index of thorn paths, thorn rods, caterpillars, thorn rings, thorn stars, Kragujevac trees, and dendrimers.


INTRODUCTION
Let G be an n-vertex simple connected graph with vertex set } ,..., , { ) (  .Clearly, . The concept of thorn graphs was introduced by GUTMAN (1998) and eventually found a variety of chemical applications; see (BYTAUTAS et al., 2001;BONCHEV and KLEIN, 2002;VUKIČEVIĆ and GRAOVAC, 2004;ZHOU, 2005;VUKIČEVIĆ et al., 2005VUKIČEVIĆ et al., , 2007;;WALIKAR et al., 2006;KLEIN et al., 2007;HEYDARI and GUTMAN, 2010;LI, 2011;ALIZADEH et al., 2014;AZARI, 2014;AZARI andIRANMANESH, 2015, 2016).The motivation for the study of thorn graphs came from a particular case, namely ) ,..., , ( , where i γ is the degree of the i-th vertex of G and γ is a constant (  A topological index is a numeric quantity that is mathematically derived in a direct and unambiguous manner from the structural graph of a molecule.It is used in theoretical chemistry for the design of chemical compounds with given physico-chemical properties or given pharmacologic and biological activities (DIUDEA, 2001).It is well known that the study of topological indices of kenograms is much more conventional than plerograms, because of their simplicity and the fact that many topological indices give highly correlated results on plerograms and kenograms (GUTMAN et al., 1998).The study of thorn graphs unifies these two approaches by giving mathematical formulae that connect the values of topological indices of kenograms and plerograms.
In this paper, we study relation between the Gutman index of a simple connected graph and its thorn graph and apply the results to compute the Gutman index of thorn paths, thorn rods, caterpillars, thorn rings, thorn stars, Kragujevac trees, and dendrimers.

DEFINITIONS AND PRELIMINARIES
In this paper, we consider connected finite graphs without any loops or multiple edges.The best known and widely used topological index is the Wiener index introduced by WIENER (1947), who used it for modeling the shape of organic molecules and for calculating several of their physico-chemical properties.The Wiener index of a graph G is defined as the sum of distances between all pairs of vertices of G, denotes the distance between the vertices u and v in G .The degree distance was introduced by DOBRYNIN and KOCHETOVA (1994) and at the same time by GUTMAN (1994) as a weighted version of the Wiener index.The degree distance of a graph G is defined as In fact, if T is a tree on n vertices, the Wiener index and degree distance are closely related by The Gutman index (also known as Schultz index of the second kind) was introduced by GUTMAN (1994) as a kind of vertex-valency-weighted sum of the distances between all pairs of vertices in a graph.Gutman revealed that in the case of acyclic structures, the index is closely related to the Wiener index and reflects precisely the same structural features of a molecular as the Wiener index does.The Gutman index of a graph G is defined as of a graph G is defined as the sum of distances between all pairs of its pendent vertices, as the sum of distances between u and all pendent vertices of G, .

RESULTS AND DISCUSSION
In this section, we establish relation between the Gutman index of a simple connected graph G and its thorn graph Ρ G , and examine several special cases of the result.
Proof.By definition of the Gutman index, we have .
By definition of the graph Ρ G , the above sum can be partitioned into four sums as follows.
The first sum 1 S consists of contriutions to The third sum 3 S is taken over all pairs of vertices such that one of them, u, is in G, and the other one, v, is in j V for It is easy to check that The fourth sum 4 S is taken over all pairs of vertices such that one of them, u, is in i V , and the other one, v, is in j V , where . Hence Eq. ( 1) is obtained by adding 1 S , 2 S , 3 S , 4 S , and simplifying the resulting expression.For every connected graph G, we define In the following theorem, we find a formula for Proof.Since for every pendent vertex i v of G, One can easily see that, S in the proof of Theorem 1.Now using Eq. ( 2), we can get Eq.( 3).As a direct consequence of Theorem 2, we get the following corollary which will be used in the next section.

Corollary 1.
Let G be a connected n-vertex graph with k pendent vertices, and let Ρ G be the thorn graph of G obtained by attaching 0 > p pendent vertices to each pendent vertex of G.
, and without loss of generality let } ,..., , { ) ( . By setting p p p p k = = = = ... , we get Eq.( 4).Now, we express some special cases of Theorem 1. , where p is a nonnegative integer.Then Corollary 3. Let G be a connected n-vertex graph with k pendent vertices, and let Ρ G be the thorn graph of G obtained by attaching 0 ≥ p pendent vertices to each pendent vertex of G.
, and without loss of generality let } ,..., , { ) ( . By setting p p p p k = = = = ...  , where , and let γ be an integer with the property , where Proof.It is easy to see that


. Now using Eq. ( 1), we can get the desired result.

APPLICATIONS
In this section, we apply the results of the previous section to compute the Gutman index of various classes of chemical graphs and nanostructures derived from thorn graphs.Let n P , n S , and n C denote the n-vertex path, star and cycle, respectively.It is easy to see that 3

Thorn paths
The thorn path k p n P , , is obtained from the path n P by adding p neighbors to each of its nonterminal vertices and k neighbors to each of its terminal vertices (see Fig. 3).Consider the path n P and choose a labeling for its vertices such that its two terminal vertices have numbers 1 and n and its nonterminal vertices have numbers .Using Eq. ( 1), we get the following theorem. .

Thorn rods
The thorn rod m n P , is a graph which includes a linear chain (termed "rod") of n vertices and degree-m terminal vertices at each of the two rod ends, where 2 ≥ m (see Fig. 4).It is easy to see that .Using Eq. ( 7), we get the following corollary. .

Thorn stars
The thorn star k p n S , , is obtained from the star n S by adding p neighbors to the center of the star and k neighbors to its terminal vertices (see Fig. 7).Consider the star n S and choose a labeling for its vertices such that its terminal vertices have numbers .Using Eq. ( 1), we get the following theorem.(see Fig. 8).Using Eq. ( 1), we get the following theorem.According to GUTMAN (2014), a Kragujevac tree T is a tree possessing a vertex of degree Substituing the above formulae in Eq. ( 11), we can get Eq.( 10).

Dendrimers
Let 0 D be the graph depicted in Fig. 11.    4) and (6), and the proof of the theorem is therefore omitted.Theorem 9. Let p and h be positive integers.Then n-tuple of nonnegative integers.The thorn graph Ρ G is the graph obtained by attaching i p pendent vertices (terminal vertices or vertices of degree one) to the vertex i v of G, for

Figure 1 .
Figure 1.The thorn graph Ρ G with parameters the vertices of Ρ G are either of degree γ or of degree one.If in addition 4 = γ , then the thorn graph Ρ G is just what CAYLEY (1874) calls a plerogram (a graph in which every atom is represented by a vertex and adjacent atoms are connected by a chemical bond) and POLYA (1937) a C-H graph.The parent graph G would then be referred to as a kenogram (CAYLEY, 1874) (a graph obtained from a plerogram by suppressing hydrogen atoms) or a C-graph (POLYA, 1937).The plerogram and kenogram of 2,3,3-trimethylpentane are depicted in Fig. 2.

Corollary 2 .
Let G be a connected n-vertex graph, and let Ρ G be the thorn graph of G with parameters p

Corollary 4 .G
Let G be a connected n-vertex graph with vertex set } be the thorn graph of G with parameters

Figure 3 .
Figure 3.The thorn path k p n P , , .

Figure 4 .
Figure 4.The thorn rod m n P , .
graph whose parent graph is the path n P and whose n nonterminal vertices are of the same degree 2 > m (see Fig.5).It is easy to see that 1

6 C
m n C , has a cycle n C as the parent, and 2 − m thorns at each cycle vertex, where 2 > m .The 3-thorn ring 3 , is depicted in Fig. 6.The m-thorn ring m n C , can be considered as the thorn graph P n C ) (, where P is the n-tuple )
vertex has number n as shown in Fig.7.Then,

Figure 7
Figure 7.The thorn star k p n S , , .
let k be any nonnegative integer.Then graph n S and choose a labeling for its vertices such that its terminal vertices star obtained by attaching i p terminal vertices to the vertex i of n

Figure 9 .
Figure 9.The rooted trees 2 B , 3 B , and k B .Their roots are indicated by large dots.

.
This vertex is said to be the central vertex of T, whereas d is the degree of T. The subgraphs of T. Recall that some (or all) branches of T may be mutually isomorphic.We denote the Kragujevac tree of degree d with branches is depicted in Fig.10.
For a fixed positive integer p , let h k denote the number of pendent vertices of AZARI and IRANMANESH, 2016), an explicit formula for computing the terminal Wiener index of the dendrimer graph h p D , was computed.Theorem 8. (AZARI and IRANMANESH, 2016) Let p and h be positive integers.The terminal Wiener index of the dendrimer graph

Table 1 .
The Gutman index of the dendrimer graphs