2012
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Note on Properties of First Zagreb Index of Graphs
2
2
Let G be a graph. The first Zagreb M1(G) of graph G is defined as: M1(G) = uV(G) deg(u)2. In this paper, we prove that each even number except 4 and 8 is a first Zagreb index of a caterpillar. Also, we show that the fist Zagreb index cannot be an odd number. Moreover, we obtain the fist Zagreb index of some graph operations.
1

1
5


M.
TAVAKOLI
Ferdowsi University of Mashhad, Iran
Ferdowsi University of Mashhad, Iran
I R Iran


F.
RAHBARNIA
Ferdowsi University of Mashhad, Iran
Ferdowsi University of Mashhad, Iran
I R Iran
Topological indices
The first and second Zagreb indices
tree
Graph operation
Strongly distancebalanced graph
Eccentric Connectivity Index of Some Dendrimer Graphs
2
2
The eccentricity connectivity index of a molecular graph G is defined as (G) = aV(G) deg(a)ε(a), where ε(a) is defined as the length of a maximal path connecting a to other vertices of G and deg(a) is degree of vertex a. Here, we compute this topological index for some infinite classes of dendrimer graphs.
1

7
18


M.
GHORBANI
Shahid Rajaee Teacher Training
University, I. R. Iran;
Shahid Rajaee Teacher Training
University,
I R Iran


KH.
MALEKJANI
Shahid Rajaee Teacher Training
University, I. R. Iran;
Shahid Rajaee Teacher Training
University,
I R Iran


A.
KHAKI
Shahid Rajaee Teacher Training
University, I. R. Iran;
Shahid Rajaee Teacher Training
University,
I R Iran
Eccentricity
topological index
Dendrimer graphs
Computing GA4 Index of Some Graph Operations
2
2
The geometricarithmetic index is another topological index was defined as 2 deg ( )deg ( ) ( ) deg ( ) deg ( ) G G uv E G G u v GA G u v , in which degree of vertex u denoted by degG (u). We now define a new version of GA index as 4 ( ) 2 ε ( )ε ( ) ( ) ε ( ) ε ( ) G G e uv E G G G u v GA G u v , where εG(u) is the eccentricity of vertex u. In this paper we compute this new topological index for two graph operations.
1

19
28


M.
SAHELI
University of Kashan, I. R. Iran
University of Kashan, I. R. Iran
I R Iran


M.
JALALI RAD
University of Kashan,
Kashan, I. R. Iran
University of Kashan,
Kashan, I. R. Iran
I R Iran
topological index
GA Index
GA_{4} index
Graph operations
On Symmetry of Some Nano Structures
2
2
It is necessary to generate the automorphism group of a chemical graph in computeraided structure elucidation. An Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix M = [dij], where for i≠j, dij is the Euclidean distance between the nuclei i and j. In this matrix dii can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce different weights for distinct nuclei. A.T. Balaban introduced some monster graphs and then M. Randic computed complexity indices of them (see A.T. Balaban, Rev. Roum. Chim. 18(1973) 841853 and M. Randic, Croat. Chem. Acta 74(3)(2001) 683 705). In this paper, we describe a simple method, by means of which it is possible to calculate the automorphism group of weighted graphs.
1

29
36


M.
GHORBANI
Shahid Rajaee Teacher Training
University, I. R. Iran
Shahid Rajaee Teacher Training
University,
I R Iran


A.
ZAEEMBASHI
Shahid Rajaee Teacher Training
University, I. R. Iran
Shahid Rajaee Teacher Training
University,
I R Iran


M.
SHAHREZAEI
Imam Hossein University,
I.R. Iran
Imam Hossein University,
I.R. Iran
I R Iran


A.
TABATABAEI ADNANI
Islamic Azad University, I. R. Iran
Islamic Azad University, I. R. Iran
I R Iran
Weighted graph
Euclidean graph
Applications of Graph Operations
2
2
In this paper, some applications of our earlier results in working with chemical graphs are presented.
1

37
43


M.
TAVAKOLI
Ferdowsi University of Mashhad, Iran
Ferdowsi University of Mashhad, Iran
I R Iran


F.
RAHBARNIA
Ferdowsi University of Mashhad, Iran
Ferdowsi University of Mashhad, Iran
I R Iran
topological index
Graph operation
Hierarchical product
Chemical graph
GeometricArithmetic Index of Hamiltonian Fullerenes
2
2
A graph that contains a Hamiltonian cycle is called a Hamiltonian graph. In this paper we compute the first and the second geometric – arithmetic indices of Hamiltonian graphs. Then we apply our results to obtain some bounds for fullerene.
1

45
50


H.
MOSTAFAEI
Islamic Azad University, Tehran, Iran
Islamic Azad University, Tehran, Iran
I R Iran


A.
ZAEEMBASHI
Shahid Rajaee Teacher Training
University, Tehran, I. R. Iran
Shahid Rajaee Teacher Training
University,
I R Iran


M.
OSTAD RAHIMI
Tehran North Branch, Islamic Azad University,
Iran
Tehran North Branch, Islamic Azad University,
I R Iran
Fullerene graphs
Hamiltonian graphs
Geometric –arithmetic index
On Counting Polynomials of Some Nanostructures
2
2
The Omega polynomial(x) was recently proposed by Diudea, based on the length of strips in given graph G. The Sadhana polynomial has been defined to evaluate the Sadhana index of a molecular graph. The PI polynomial is another molecular descriptor. In this paper we compute these three polynomials for some infinite classes of nanostructures.
1

51
58


M.
GHORBANI
Shahid Rajaee Teacher Training
University, I. R. Iran
Shahid Rajaee Teacher Training
University,
I R Iran


M.
SONGHORI
Shahid Rajaee Teacher Training
University, I. R. Iran
Shahid Rajaee Teacher Training
University,
I R Iran
Omega polynomial
PI polynomial
Nanostar dendrimers
Computing Chemical Properties of Molecules by Graphs and Rank Polynomials
2
2
The topological index of a graph G is a numeric quantity related to G which is invariant under automorphisms of G. The Tutte polynomial of is a polynomial in two variables defined for every undirected graph contains information about connectivity of the graph. The PadmakarIvan, vertex PadmakarIvan polynomials of a graph are polynomials in one variable defined for every simple connected graphs that are undirected. In this paper, we compute these polynomials of two infinite classes of dendrimer nanostars.
1

59
65


M.
MOGHARRAB
Persian Gulf University,
I.R. Iran
Persian Gulf University,
I.R. Iran
I R Iran


G.
FATHTABAR
University of
Kashan, I. R. Iran
University of
Kashan, I. R. Iran
I R Iran
Dendrimers
Tutte polynomial
PIpolynomial
A Note on Atom Bond Connectivity Index
2
2
The atom bond connectivity index of a graph is a new topological index was defined by E. Estrada as ABC(G) uvE (dG(u) dG(v) 2) / dG(u)dG(v) , where G d ( u ) denotes degree of vertex u. In this paper we present some bounds of this new topological index.
1

67
75


S.
HEIDARI RAD
Shahid Rajaee Teacher Training
University, I. R. Iran
Shahid Rajaee Teacher Training
University,
I R Iran


A.
KHAKI
Shahid Rajaee Teacher Training
University, I. R. Iran
Shahid Rajaee Teacher Training
University,
I R Iran
topological index
ABC Index
Nanotube
Nanotori