Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 583 https://internationalpubls.com On Graph Polynomials and Their Applications of Nanostar Structures Mohammed W. Mihan 1 , Habib Azanchiler 2 , Nabeel E. Arif 3 , Sara Eslameian 4 1 Department of Mathematics , Faculty of sciences , Urmia University, Iran, Email : iraq7100@yahoo.com 2 Department of Mathematics , Faculty of sciences , Urmia University, Iran, 3 Department of Mathematics, College of Computer sciences and Mathematics , Tikrit University, iraq , Email : nabarif@tu.edu.iq 4 Department of Mathematics , Faculty of sciences , Urmia University, Iran Article History: Received: 04-06-2024 Revised: 03-07-2024 Accepted: 30-07-2024 Abstract: A Nanostars (nanostructure) refers to an entity that occupies an intermediate scale between microscopic and atomic structures. In this paper will used the nanostructure for two types of chemical components known as the fullerene dendrimer is represented as ,and Polypropylenimine octaamine dendrimer known as to get new form to redefine the third Zagrebpolynomial ,general Randicpolynomial , general sum- connectivity polynomial , fourth Zagreb, fifth Zagreb, and harmonic polynomials. Keywords: Graph Polynomials, chemical graph, Nanostars, Dendrmiers. 1. Introduction Mathematical chemistry is a specialised field within the realm of chemistry that employs mathematical theories and concepts to analyses and explain chemical structures[1]. Chemical graph theory is a subfield of mathematical chemistry that establishes a connection between graph theory and chemical structures. The concept of the chemical graph emerged throughout the eighteenth century as a result of the intellectual contributions of Isaac Newton. John Dalton, in 1805, introduced the initial model for representing different types of atoms using distinct circles[2][3]. A Nanostars (nanostructure) refers to an entity that occupies an intermediate scale between microscopic and atomic structures. This phenomenon occurs as a result of a physical measurement that is less than 100 nanometers [4]. The first synthesis of dendrimers was accomplished by Fritz Vogtle in 1978 , who employed several synthetic techniques[5]. These approaches included the contributions of RG Denkewalter and Donald Tomalia in 1980[6][7]. In1990[8] George R. Newkome, Craig Hawker, and Jean Frechet proposed a fusion synthesis technique. The prevalence of dendrimers has experienced a significant surge. Prior to 2005, a substantial number of scholarly articles and patents, exceeding 5000 papers, were produced[9]. Consider a graph G that is both simple and linked ( molecular graph),The symbols V(G) and E(G) denote the vertex set and edge set of graph G, respectively. Let v be any vertex in the set V (G). We define Gd (v ) as the degree of vertex v and N(v) as the set of vertices that are neighbours of v, such Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 584 https://internationalpubls.com that GN (v ) d (v ) . The 1Z ( G ) and 2Z ( G ) indices, also referred to as the first and second Zagreb indices, these indices were introduced in a publication in 1971 [10]. Also in (2011) Fath –Tabar [11] were defined the third Zagreb polynomial u vd d 3 vu E ( G ) Z ( G ) x     In ( 2022) P.Gladyis et. al [12] used the fourth and fifth Zagreb polynomials of nanostar dendrimer Dn , which are depends on cutting number of the vertices v v ud ( d d ) 4 vu E ( G ) Z ( G ) x     u v ud ( d d ) 5 vu E ( G ) Z ( G ) x     In (2021) Abdul Jalil M. Khalaf [13] used some polynomials to compute cellulose's chemical structure like as The general sum-connectivity polynomial [dv du] vu E(G) (G, x) x       Randic polynomial [ dv du ] vu E (G) R ( G ,x ) x       In (2018) Juan C. Hernández-Gómezused et. al [14], used harmonic polynomial H(G) to obtain several properties of the harmonic polynomial to obtain omany classical symmetric operations of graphsand . In (2020) ALI AHMAD et. al [15], compute any degree-based topological polynomials for Optical Transpose Interconnection System swapped network.The harmonic polynomial is given as v ud d 1 vuE ( G ) H ( G ) x     Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 585 https://internationalpubls.com 2. Main Results: 2.1 First Result (Fullerene dendrimer) : Fig 1. Fullerene dendrimer 1NS [ i ] . In (Fig. 1), the fullerene dendrimer is nanostar represented as 1NS [i] , where n is the number of steps of growth, as illustrated. The molecular graph of 1NS [i] has two branches in symmetrical arrangement contain six types of degrees of the end vertices (1,3), (2,2), (2,3), (3,3), (3,4), and (4,4) . Hence, by doing a direct calculation, we obtain     1 1 13 v uE ( NS [i]) e { e E :d 1,d 3 }     2 1 22 v uE ( NS [i]) e { e E :d 2 ,d 2 }     3 1 23 v uE ( NS [i]) e { e E :d 2 ,d 3 }     4 1 33 v uE ( NS [i]) e { e E :d 3 ,d 3 }     5 1 34 v uE ( NS [i]) e { e E :d 3 ,d 4 }     6 1 44 v uE ( NS [i]) e { e E :d 4 ,d 4 }  i 1 13e 2 ,  i 1 22e 2 2 ,   i 1 23e 32 2 8 , 33e 86 , 34e 6 , and 44e 3 . Table 1: the value of degree in 1NS [i] where d (v ,u ) = (1,3), (2,2), (2,3), (3,3),(3,4), and (4,4) . stage degree i=1 i=2 i=3 d(1,3) 2 4 8 d(2,2) 4 6 10 d(2,3) 8 24 56 d(3,3) 86 86 86 d(3,4) 6 6 6 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 586 https://internationalpubls.com d(4,4) 3 3 3 Theorem 1: Let 1NS [i] be the nanostar with i={0,1,2,…} the redefine third Zagreb polynomial is            i 1 12 i 1 16 i 1 30 3 1 54 84 128 ReZ G(NS [i],x) 2 x ( 2 2 )x ( 32 2 8 )x 86 x 6 x 3x Proof: in (Fig.1) we can see there are two similar branches depends on the degree of the end vertices that has six types of end degree we refers to it by (table 1) so by using the definition we obtain 1 (dv du)(dv du) 3 1 vu E(NS [i]) Re Z G(NS [i], x) x                                    1 1 1 1 1 1 ( 1 3 )( 1 3 ) ( 2 2 )( 2 2 ) vu E(NS [i]) vu E(NS [i]) ( 2 3 )( 2 3 ) ( 3 3 )( 3 3 ) vu E(NS [i]) vu E(NS [i]) ( 3 4 )( 3 4 ) ( 4 4 )( 4 4 ) vu E(NS [i]) vu E(NS [i]) x x x x x x            i 1 12 i 1 16 i 1 30 3 1 54 84 128 ReZ G(NS [i],x) 2 x ( 2 2 )x ( 32 2 8 )x 86 x 6 x 3x Theorem 2: Let 1NS [i] be the nanostar with i={0,1,2,…} and  is positive integer the general sum-connectivity polynomial is      i 1 ( 4 ) ( 6 ) (7 ) ( 8 ) 1(NS [i],x ) ( 2 2 2 )x 86x 6x 3x      Proof: in (Fig.1) we can see there are two similar branches depends on the degree of the end vertices that has six types of end degree we refers to it by (table 1) so by using the definition to the general sum-connectivity polynomial we obtain 1 [dv du] 1 vu E(NS [i]) (NS [i], x) x                               1 1 1 1 1 1 ( 1 3 ) ( 2 2 ) vu E(NS [i]) vu E(NS [i]) ( 2 3 ) ( 3 3 ) vu E(NS [i]) vu E(NS [i]) ( 3 4 ) ( 4 4 ) vu E(NS [i]) vu E(NS [i]) x x x x x x       Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 587 https://internationalpubls.com             ( 1 3 ) ( 2 2 ) ( 2 3 ) 1 1 2 1 3 1 ( 3 3 ) ( 3 4 ) ( 4 4 ) 4 1 5 1 6 1 E ( NS [i] x E ( NS [i] x E ( NS [i] x E ( NS [i] x E ( NS [i] x E ( NS [i] x             i 1 ( 4 ) i 1 ( 4 ) ( 6 ) (7 ) ( 8 )2 x ( 2 2 )x 86x 6x 3x           i 1 ( 4 ) ( 6 ) (7 ) ( 8 )( 2 2 2 )x 86x 6x 3x     Theorem 3: Let 1NS [i] be the nanostar with i={0,1,2,…} and  is positive integer the general Randic polynomial is            i 1 ( 3 ) i 1 ( 4 ) i 1 ( 6 ) 1 ( 9 ) ( 12 ) ( 16 ) R (NS [i],x ) 2 x ( 2 2 )x ( 32 2 8 )x 86 x 4x 3x        Proof: in (Fig.1) we can see there are two similar branches depends on the degree of the end vertices that has six types of end degree we refers to it by (Table 1) so by using the definition the general Randic polynomial     1 [ dv du ] 1 vu E(NS [i]) R (NS [i],x ) x                           1 1 1 1 1 1 ( 1 3 ) ( 2 2 ) ( 2 3 ) vu E(NS [i]) vu E(NS [i]) vu E(NS [i]) ( 3 3 ) ( 3 4 ) ( 4 4 ) vu E(NS [i]) vu E(NS [i]) vu E(NS [i]) x x x x x x                   ( 1 3 ) ( 2 2 ) ( 2 3 ) 1 1 2 1 3 1 ( 3 3 ) ( 3 4 ) ( 4 4 ) 4 1 5 1 6 1 E ( NS [i] x E ( NS [i] x E ( NS [i] x E ( NS [i] x E ( NS [i] x E ( NS [i] x                  i 1 ( 3 ) i 1 ( 4 ) i 1 ( 6 ) ( 9 ) ( 12 ) ( 16 ) 2 x ( 2 2 )x ( 32 2 8 )x 86 x 4x 3x       Theorem 4: Let 1NS [i] be the nanostar with i={0,1,2,…} the fourth Zagreb polynomial is            i 1 4 i 1 8 i 1 10 4 1 18 21 32 Z (NS [i],x ) 2 x ( 2 2 )x ( 32 2 8 )x 86 x 6 x 3x Proof :in (Fig.1) we can see there are two similar branches depends on the degree of the end vertices that has six types of end degree we refers to it by (table 1) so by using the definition to the fourth Zagreb polynomial     v v u 1 d ( d d ) 4 1 vu E(NS [i]) Z (NS [i],x ) x Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 588 https://internationalpubls.com                         1 1 1 1 1 1 ( 1 3 ) 2( 2 2 ) vu E(NS [i]) vu E(NS [i]) 2( 2 3 ) 3( 3 3 ) vu E(NS [i]) vu E(NS [i]) 3( 3 4 ) 4( 4 4 ) vu E(NS [i]) vu E(NS [i]) x x x x x x       4 8 10 1 4 2 4 3 4 18 21 32 4 4 5 4 6 4 E ( NS [i] x E ( NS [i] x E ( NS [i] x E ( NS [i] x E ( NS [i] x E ( NS [i] x            i 1 4 i 1 8 i 1 10 18 21 32 2 x ( 2 2 )x ( 32 2 8 )x 86 x 6 x 3x Theorem 5: Let 1NS [i] be the nanostar when i={0,1,2,…} the fifth Zagreb polynomial is            i 1 8 i 1 12 i 1 15 5 1 18 28 32 Z (NS [i],x ) ( 2 2 )x ( 2 )x ( 32 2 8 )x 86 x 6 x 3x Proof : in (Fig.1) we can see there are two similar branches depends on the degree of the end vertices that has six types of end degree we refers to it by (table 1) so by using the definition to the fifth Zagreb polynomial     u v u 1 d ( d d ) 5 1 uv E(NS [i]) Z (NS [i],x ) x                         1 1 1 1 1 1 3( 1 3 ) 2( 2 2 ) vu E(NS [i]) vu E(NS [i]) 3( 2 3 ) 3( 3 3 ) vu E(NS [i]) vu E(NS [i]) 4( 3 4 ) 4( 4 4 ) vu E(NS [i]) vu E(NS [i]) x x x x x x             3( 1 3 ) 2( 2 2 ) 3( 2 3 ) 1 1 2 1 3 1 3( 3 3 ) 4( 3 4 ) 4( 4 4 ) 4 1 5 1 6 1 E ( NS [i] x E ( NS [i] x E ( NS [i] x E ( NS [i] x E ( NS [i] x E ( NS [i] x            i 1 12 i 1 8 i 1 15 18 48 32 2 x ( 2 2 )x ( 32 2 8 )x 86 x 6 x 3x            i 1 8 i 1 12 i 1 15 18 28 32 ( 2 2 )x ( 2 )x ( 32 2 8 )x 86 x 6 x 3x Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 589 https://internationalpubls.com Theorem 6: Let 1NS [i] be the nanostar with i={0,1,2,…} by using the harmonic polynomial is           i 1 3 i 1 4 5 6 7 ( 2 2 2 )x ( 32 2 8 )x 86 x 6 x 3x Proof : in (Fig.1) we can see there are two similar branches depends on the degree of the end vertices that has six types of end degree we refers to it by (table 1) so by using the definition to the harmonic polynomial v ud d 1 vu E ( x ) H ( G ,x ) x                                    1 1 1 1 1 1 ( 1 3 ) 1 ( 2 2 ) 1 vu E(NS [i]) vu E(NS [i]) ( 2 3 ) 1 ( 3 3 ) 1 vu E(NS [i]) vu E(NS [i]) ( 3 4 ) 1 ( 4 4 ) 1 vu E(NS [i]) vu E(NS [i]) x x x x x x       3 3 4 1 1 2 1 3 1 5 6 7 4 1 5 1 6 1 E ( NS [i] x E ( NS [i] x E ( NS [i] x E ( NS [i] x E ( NS [i] x E ( NS [i] x           i 1 3 i 1 4 5 6 7 ( 2 2 2 )x ( 32 2 8 )x 86 x 6 x 3x Fig.2 : Comparison between 3 rd Zagreb (green), sum conn. (blue), 4 th Zageb (black), 5 th Zagreb (white ), and harmonic (brown) plolynomials . 2.2 Second Results : In this section we will study all polynomials that we remember in the (First Result) by using the another chemical setructuer in (Fig.3) known as (Polypropylenimine octaamine dendrimer ) will be denoted as 2NS [i] .where n is the number of steps of growth, as illustrated. The molecular graph of 2NS [i] has three types of degrees of the end vertices (1,2), (2,2),and (2,3) . In symmetrical arrangement with four analogous branches. Hence, by doing a direct calculation, we obtain     1 2 12 v uE ( NS [i]) e { e E :d 1,d 2 } Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 590 https://internationalpubls.com     2 2 22 v uE ( NS [i]) e { e E :d 2 ,d 2 }     3 2 23 v uE ( NS [i]) e { e E :d 2 ,d 3 }       i 1 i i 12 22 23e 2 ,e (8 2 5 ),e (6 2 6 ) Figure 3. Polypropylenimine octaamine dendrimer 2NS [i] Table 2: the value of degree in 2NS [p] where d (v ,u ) = (1,2), (2,2), and (2,3) with stage i={0,1,2,…} stage degree i=o i=1 i=2 d(1,2) 2 4 8 d(2,2) 3 11 27 d(2,3) 0 6 18 Theorem 1: Let 2NS [i] be the nanostar with i={0,1,2,…} the redefine third Zagreb polynomial is       i 1 6 i 16 i 30 3 2RZ G(NS [i],x) 2 x (8 2 5 )x (6 2 6 )x Proof: in (Fig.3) we can see there are two similar branches depends on the degree of the end vertices that has three types of end degree we refers to it by (table 2) so by using the definition to redefine the third Zagreb polynomial 2 (dv du)(dv du) 3 2 vu E(NS [i]) Re Z G(NS [i], x) x                    2 2 2 ( 1 2 )( 1 2 ) ( 2 2 )( 2 2 ) ( 2 3 )( 2 3 ) vu E(NS [i]) vu E(NS [i]) vu E(NS [i]) x x x Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 591 https://internationalpubls.com          2 2 2 6 16 1 2 1 2 vu E(NS [i]) vu E(NS [i]) 30 1 2 vu E(NS [i]) E ( NS [i] x E ( NS [i] x E ( NS [i] x       i 1 6 i 16 i 302 x (8 2 5 )x (6 2 6 )x Theorem 2: Let 2NS [i] be the nanostar with i={0,1,2,…} and  is positive integer the general sum-connectivity polynomial is       i 1 ( 3 ) i ( 4 ) i ( 5 ) 2( NS [i],x) 2 x ( 8 2 5 )x (6 2 6 )x     Proof : in (Fig.3) we can see there are two similar branches depends on the degree of the end vertices that has three types of end degree we refers to it by (table 2) so by using the definition the general sum-connectivity polynomial then we obtain 2 [dv du] 2 vu E(NS [i]) (NS [i], x) x                   2 2 2 ( 1 2 ) ( 2 2 ) vu E(NS [i]) vu E(NS [i]) ( 2 3 ) vu E(NS [i]) x x x        ( 1 2 ) ( 2 2 ) ( 2 3 ) 1 2 2 2 3 2E ( NS [i] x E ( NS [i] x E ( NS [i] x          i 1 ( 3 ) i ( 4 ) i ( 5 )2 x ( 8 2 5 )x (6 2 6 )x    Theorem 3: Let 2NS [i] be the nanostar with i={0,1,2,…} and  is positive integer the general Randic polynomial is       i 1 ( 2 ) i ( 4 ) i ( 6 ) 2R (NS [i],x ) 2 x ( 8 2 5 )x (6 2 6 )x     Proof: in (Fig.3) we can see there are two similar branches depends on the degree of the end vertices that has three types of end degree we refers to it by (table 2) so by using the definition to the general Randic polynomial     2 [ dv du ] 2 vu E(NS [i]) R (NS [i],x ) x              2 2 2 ( 1 2 ) ( 2 2 ) ( 2 3 ) vu E(NS [i]) vu E(NS [i]) vu E(NS [i]) x x x        ( 1 2 ) ( 2 2 ) ( 2 3 ) 1 2 2 2 3 2E ( NS [i] x E ( NS [i] x E ( NS [i] x          i 1 ( 2 ) i ( 4 ) i ( 6 )2 x ( 8 2 5 )x (6 2 6 )x    Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 592 https://internationalpubls.com Theorem 5: Let 2NS [i] be the nanostar with i={0,1,2,…} the fourth Zagreb polynomial is       i 1 3 i 8 i 10 4 2Z (NS [i],x) 2 x (8 2 5 )x (6 2 6 )x Proof : in (Fig.3) we can see there are two similar branches depends on the degree of the end vertices that has three types of end degree we refers to it by (table 2) so by using definition to the fourth Zagreb polynomial     v v u 2 d ( d d ) 4 2 vu E(NS [i]) Z (NS [i],x ) x             2 2 2 ( 1 2 ) 2( 2 2 ) vu E(NS [i]) vu E(NS [i]) 2( 2 3 ) vu E(NS [i]) x x x   3 8 10 1 2 2 2 3 2E ( NS [i] x E ( NS [i] x E ( NS [i] x       i 1 3 i 8 i 102 x (8 2 5 )x (6 2 6 )x Theorem 6: Let 2NS [i] be the nanostar with i={0,1,2,…} the fifth Zagreb polynomial is       i 1 3 i 8 i 10 5 2Z (NS [i],x) 2 x (8 2 5 )x (6 2 6 )x Proof : in (Fig.3) we can see there are two similar branches depends on the degree of the end vertices that has three types of end degree we refers to it by (table 2) so by using definition to the fifth Zagreb polynomial     u v u 2 d ( d d ) 5 2 uv E(NS [i]) Z (NS [i],x ) x             2 2 2 2( 1 2 ) 2( 2 2 ) vu E(NS [i]) vu E(NS [i]) 3( 2 3 ) vu E(NS [i]) x x x   6 8 15 1 2 2 2 3 2E ( NS [i] x E ( NS [i] x E ( NS [i] x       i 1 6 i 8 i 152 x (8 2 5 )x (6 2 6 )x Theorem 7: Let 2NS [i] be the nanostar with i={0,1,2,…} the harmonic polynomial is       i 1 2 i 3 i 4 2H(NS [i],x) 2 x (8 2 5 )x (6 2 6 )x Proof : in (Fig.3) we can see there are two similar branches depends on the degree of the end vertices that has three types of end degree we refers to it by (table 2) so by using definition to the harmonic polynomial Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 593 https://internationalpubls.com      v u 2 d d 1 2 uv E(NS [i]) H(NS [i],x ) x                2 2 2 ( 1 2 ) 1 ( 2 2 ) 1 vu E(NS [i]) vu E(NS [i]) ( 2 3 ) 1 vu E(NS [ i ]) x x x   2 3 4 1 2 2 2 3 2E ( NS [i] x E ( NS [i] x E ( NS [i] x       i 1 2 i 3 i 42 x (8 2 5 )x (6 2 6 )x . 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