First and Second Zagreb Eccen icity I dices of Thorny Graphs

The Zagreb eccentricity indices are the eccentricity reformulation of the Zagreb indices. Let H be a simple graph. The first Zagreb eccentricity index (E1(H)) is defined to be the summation of squares of the eccentricity of vertices, i.e., E1(H) = ∑u∈V(H) Symmetry 2016, 9, 7; doi: 10.3390/sym9010007 www.mdpi.com/journal/symmetry Article First and Second Zagreb Eccentricity Indices of Thorny Graphs Nazeran Idrees 1,*, Muhammad Jawwad Saif 2, Asia Rauf 3 and Saba Mustafa 1 1 Department of Mathematics, Government College University Faisalabad, 38000 Faisalabad, Pakistan; sabamustafa48@gmail.com (S.M.) 2 Department of Applied Chemistry, Government College University Faisalabad, 38000 Faisalabad, Pakistan; jawwadsaif@gmail.com (M.J.S.); 3 Department of Mathematics, Government College Women University Faisalabad, 38000 Faisalabad, Pakistan; Asia.rauf@gmail.com (A.R.) * Corresponding author: nazeranjawwad@gmail.com; Academic Editor: Angel Garrido Received: 21 November 2016; Accepted: 22 December 2016; Published: date Abstr ct: Th Zagreb eccentricity indices are the ecc nt icity reformulation of the Zagreb indices. Let be a simple graph. The first Zag eb eccentricity index ( ) is defined to be the summation of squares of the eccentricity of vert c s, i.e., ∑ Ɛ ∈ . The second Zagreb eccentricity index ( ) is the summation of product of the eccentricities of the adjacent vertices, i.e., ∑ Ɛ Ɛ ∈ . We obtain the thorny graph of a graph by attaching thorns i.e., vertices of degree one to every vertex of . In this paper, we will find closed formulation for the first Zagreb eccentricity index and second Zagreb eccentricity index of different well known classes of thorny graphs.


Introduction
In theoretical chemistry, molecular descriptors or topological indices are utilized to configure properties of chemical compounds.A topological index is a real number connected with chemical structure indicating relationships of chemical configuration with different physical properties, chemical reactivity or biological activity, which is utilized to understand properties of chemical compounds in theoretical chemistry.Topological indices have been observed to be helpful in chemical documentation, isomer discrimination, structure-property relations, structure-activity (SAR) relations and pharmaceutical medication plans.All through the paper, all graphs are considered to be simple and connected.
Let ) is defined to be the summation he eccentricity of vertices, i.e., ∑ Ɛ ∈ .The second Zagreb eccentricity is the summation of product of the eccentricities of the adjacent vertices, i.e., Ɛ . We obtain the thorny graph of a graph by attaching thorns i.e., vertices of every vertex of .In this paper, we will find closed formulation for the first Zagreb dex and second Zagreb eccentricity index of different well known classes of thorny aphs; vertices; complete graph; path; star; cycle ical chemistry, molecular descriptors or topological indices are utilized to configure hemical compounds.A topological index is a real number connected with chemical ating relationships of chemical configuration with different physical properties, ivity or biological activity, which is utilized to understand properties of chemical theoretical chemistry.Topological indices have been observed to be helpful in mentation, isomer discrimination, structure-property relations, structure-activity s and pharmaceutical medication plans.All through the paper, all graphs are e simple and connected.
,  Several topological indices depend upon the eccentricity of the vertices and are very effective in drug design.Sharma, Goswami and Madan [2] proposed the eccentric connectivity index of the graph H, which is defined as .
Recently, the first Zagreb eccentricity index and second Zagreb eccentricity index E 1 and E 2 have been proposed as the revised versions of the Zagreb indices M 1 and M 2 , respectively, by Ghorbani and Hosseinzadeh [10].The first Zagreb eccentricity index (E 1 ) and the second Zagreb eccentricity index (E 2 ) of a graph H are defined as

Introduction
In theoretical chemistry, molecular descriptors or topological indices are utilized to configure properties of chemical compounds.A topological index is a real number connected with chemical structure indicating relationships of chemical configuration with different physical properties, chemical reactivity or biological activity, which is utilized to understand properties of chemical compounds in theoretical chemistry.Topological indices have been observed to be helpful in chemical documentation, isomer discrimination, structure-property relations, structure-activity (SAR) relations and pharmaceutical medication plans.All through the paper, all graphs are considered to be simple and connected. Let

Introduction
In theoretical chemistry, molecular descriptors or topological indices are utilized to configure properties of chemical compounds.A topological index is a real number connected with chemical structure indicating relationships of chemical configuration with different physical properties, chemical reactivity or biological activity, which is utilized to understand properties of chemical compounds in theoretical chemistry.Topological indices have been observed to be helpful in chemical documentation, isomer discrimination, structure-property relations, structure-activity (SAR) relations and pharmaceutical medication plans.All through the paper, all graphs are considered to be simple and connected.
Let ) is the summation of product of the eccentricities of the adjacent vertices, i.e., ∑ Ɛ Ɛ

Introduction
In theoretical chemistry, molecular descriptors or topological indices are utilized to configure properties of chemical compounds.A topological index is a real number connected with chemical structure indicating relationships of chemical configuration with different physical properties, chemical reactivity or biological activity, which is utilized to understand properties of chemical compounds in theoretical chemistry.Topological indices have been observed to be helpful in chemical documentation, isomer discrimination, structure-property relations, structure-activity (SAR) relations and pharmaceutical medication plans.All through the paper, all graphs are considered to be simple and connected.respectively.Das et al. [11] gave a few lower and upper bounds on the first Zagreb eccentricity index and the second Zagreb eccentricity index of trees and graphs, and also characterized the extremal graphs.Nilanjan [12] computed a few new lower and upper bounds on the first Zagreb eccentricity index and the second Zagreb eccentricity index.Zhaoyang and Jianliang [13] computed Zagreb eccentricity indices under different graph operations.Farahani [14] computed precise equations for the First Zagreb Eccentricity index of Polycyclic Aromatic Hydrocarbons.Evidently, Zagreb indices and the family of all connectivity indices express mathematically attractive invariants.In this manner, we expect numerous more studies on these indices and anticipate further development of this area of mathematical chemistry.

Results and Discussion
Consider a graph H with vertex set {u 1 , u 2 , . . . ,u m } and a set of positive integers {p 1 , p 2 , . . . ,p m }.The thorn graph of H, denoted by H * (p 1 , p 2 , . . . ,p n ), is obtained by attaching p j pendant vertices to u j for each j.The idea of a thorn graph was presented by Gutman [15], and various studies on thorn graphs and different topological indices have been conducted by some researchers in the recent past [16][17][18][19].In this paper, we will derive explicit expressions for computing the first Zagreb eccentricity index and the second Zagreb eccentricity index of thorny graphs of some well-known classes of graphs like complete graphs, complete bipartite graphs, star graphs, cycles and paths.

The Thorny Complete Graph
Suppose that we take the complete graph K m with m vertices.Obviously,

2
. The thorny complete graph K * m is obtained from K m by attaching p j thorns at each vertex of K m , j = 1, 2, . . ., m. Suppose that the total number of thorns attached to K m are denoted by T.
Theorem 1.The first Zagreb eccentricity index and the second Zagreb eccentricity index of K * m are given by: 2

The Thorny Complete Bipartite Graph
Assume that we take the complete bipartite graph K n,m having (n + m) vertices.Obviously, the eccentricities are equal to two for all the vertices of K n,m .Then, E 1 (K n,m ) = 4(n + m) and E 2 (K n,m ) = 4nm.The thorny complete bipartite graph K * n,m is attained by attaching pendant vertices to each vertex of K n,m .Let T be the total number of pendent vertices.
Theorem 2. The first Zagreb eccentricity index and the second Zagreb eccentricity index of K * n,m are given by: Proof.Suppose that {v 1 , v 2 , . . . ,v n , u 1 , u 2 , . . . ,u m } is the vertex set of K n,m , and let v ik be the newly attached pendant vertices to v i , i = 1, 2, . . ., n; k = 1, 2, . . ., p i and u jl be the pendant vertices of u j , j = 1, 2, . . ., m; k = 1, 2, . . ., p l .Then, the eccentricity of the vertices of K * n,m is given by ricity Indices of Thorny Graphs

Introduction
In theoretical chemistry, molecular descriptors or topological indices are utilized to conf properties of chemical compounds.A topological index is a real number connected with chem structure indicating relationships of chemical configuration with different physical prope chemical reactivity or biological activity, which is utilized to understand properties of chem compounds in theoretical chemistry.Topological indices have been observed to be helpf chemical documentation, isomer discrimination, structure-property relations, structure-act (SAR) relations and pharmaceutical medication plans.All through the paper, all graphs considered to be simple and connected.

Introduction
In theoretical chemistry, molecular descriptors or topological indices are utilized to configure properties of chemical compounds.A topological index is a real number connected with chemical structure indicating relationships of chemical configuration with different physical properties, chemical reactivity or biological activity, which is utilized to understand properties of chemical compounds in theoretical chemistry.Topological indices have been observed to be helpful in chemical documentation, isomer discrimination, structure-property relations, structure-activity (SAR) relations and pharmaceutical medication plans.All through the paper, all graphs are considered to be simple and connected..J.S.); ics, Government College Women University Faisalabad, 38000 Faisalabad, l.com (A.R.) zeranjawwad@gmail.com; ) is defined to be the summation of squares of the eccentricity of vertices, i.e., ∑ Ɛ ∈ .The second Zagreb eccentricity index ( ) is the summation of product of the eccentricities of the adjacent vertices, i.e., ∑ Ɛ Ɛ

Introduction
In theoretical chemistry, molecular descriptors or topological indices are utilized to configure properties of chemical compounds.A topological index is a real number connected with chemical structure indicating relationships of chemical configuration with different physical properties, chemical reactivity or biological activity, which is utilized to understand properties of chemical compounds in theoretical chemistry.Topological indices have been observed to be helpful in chemical documentation, isomer discrimination, structure-property relations, structure-activity (SAR) relations and pharmaceutical medication plans.All through the paper, all graphs are considered to be simple and connected.
from which we get the desired result.Now, and the result follows.

The Thorny Cycle
Let C m be a cycle having m vertices and m edges.Clearly, , i f m is odd m(m+2) 2 +T(m+4) 2  4   , i f m is even and , i f m is even , respectively.
Thus, when m is an odd number, the first Zagreb eccentricity index of Now, when m is an even number, the first Zagreb eccentricity index of C * m is given by Symmetry 2017, 9, 7 6 of 9 Next, we proceed for the second Zagreb eccentricity index as T.

The Thorny Path Graph
Consider the path graph P m with m vertices.If m is even, then we write m = 2n + 2, and suppose that the vertices of P m are serially indicated by v , where the centers of the path P 2n+2 are v 0 and v 0 having eccentricity n + 1.If m is odd, then we write m = 2n + 1, and we suppose that we have v n , v n−1 , . . ., v 2 , v 1 , v 0 , v 1 , v 2 , . . ., v n−1 , v n as the consecutive vertices of P m , where the center of the path P 2n+1 is v 0 having the eccentricity n.Then, the thorny path graph P * m is obtained from P m by attaching p j and p j pendant vertices to each v j and v j (j = 1, 2, . . ., n), respectively.We define p 0 = 0. Now, we will find the first Zagreb eccentricity index and the second Zagreb eccentricity index of P * m .
Theorem 5.The first Zagreb eccentricity index and the second Zagreb eccentricity index of P * m are given by , and Proof.If m = 2n + 2, then all the vertices of P * m have eccentricities ε , . . ., p j .Thus, the Zagreb eccentricity indices of P * m are given by Symmetry 2017, 9, 7 7 of 9 and In addition, (p j + p j )(n + j + 1)(n + j + 2), and we obtain the equality.

Conclusions
In this article we computed closed formulas for computing first Zagreb eccentricity index as well as second Zagreb eccentricity index for thorny graphs of important families of graphs like complete graph, complete bipartite graph, cycle, star and path.These relations are given in Theorems 1-4.Moreover, it can be observed from these formulas that values of these indices increase by increasing the number of vertices and number of thorns attached to graphs.These invariants have applications in computational chemistry.

, 2 H
be a simple graph with | | vertices and | | edges.For ∈ , degree of u, denoted by , is number of vertices attached to in the graph.The maximum distance from a vertex to any other vertex in the graph is called eccentricity of the vertex and is denoted by Ɛ i.e., Ɛ max , | ∈ , where , denotes the distance between and in .The first Zagreb index ( ) and second Zagreb index are the oldest known indices introduced by Gutman and Trinajstić [1] defined as (u).The second Zagreb eccentricity index (E 2 (H)) is the summation of product of the eccentricities of the adjacent vertices, i.e., E 2 (H) = ∑ uv∈E(H) College University Faisalabad, 38000 Faisalabad, Pakistan; ail.com (S.M.) lied Chemistry, Government College University Faisalabad, 38000 Faisalabad, Pakistan; com (M.J.S.); hematics, Government College Women University Faisalabad, 38000 Faisalabad, @gmail.com(A.R.) hor: nazeranjawwad@gmail.com; el Garrido er 2016; Accepted: 22 December 2016; Published: date eb eccentricity indices are the eccentricity reformulation of the Zagreb indices.aph.The first Zagreb eccentricity index ( ) is defined to be the summation entricity of vertices, i.e., ∑ Ɛ Zagreb eccentricity indices are the eccentricity reformulation of the Zagreb indices.ple graph.The first Zagreb eccentricity index (

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be a simple graph with | | vertices and | | edges.For ∈ , degree y , is number of vertices attached to in the graph.The maximum distance from other vertex in the graph is called eccentricity of the vertex and is denoted by Ɛ ax , | ∈ , where , denotes the distance between and in .The first ) and second Zagreb index are the oldest known indices introduced by Gutman 1] defined as In theoretical chemistry, molecular descriptors or topological indices are utilized to configure properties of chemical compounds.A topological index is a real number connected with chemical structure indicating relationships of chemical configuration with different physical properties, chemical reactivity or biological activity, which is utilized to understand properties of chemical compounds in theoretical chemistry.Topological indices have been observed to be helpful in chemical documentation, isomer discrimination, structure-property relations, structure-activity (SAR) relations and pharmaceutical medication plans.All through the paper, all graphs are considered to be simple and connected.Let H = (V, E) be a simple graph with m = |V| vertices and n = |E| edges.For u ∈ V, degree of u, denoted by d(u), is number of vertices attached to u in the graph.The maximum distance from a vertex to any other vertex in the graph H is called eccentricity of the vertex and is denoted by 1. Introduction In theoretical chemistry, molecular descriptors or topological indices are utilize properties of chemical compounds.A topological index is a real number connected structure indicating relationships of chemical configuration with different physi H College University Faisalabad, 38000 Faisalabad, Pakistan; nment College University Faisalabad, 38000 Faisalabad, Pakistan; College Women University Faisalabad, 38000 Faisalabad, mail.com;ecember 2016; Published: date es are the eccentricity reformulation of the Zagreb indices.b eccentricity index ( ) is defined to be the summation i.e., ∑ Ɛ The second Zagreb eccentricity ct of the eccentricities of the adjacent vertices, i.e., rny graph of a graph by attaching thorns i.e., vertices of paper, we will find closed formulation for the first Zagreb ccentricity index of different well known classes of thorny raph; path; star; cycle descriptors or topological indices are utilized to configure pological index is a real number connected with chemical H (u) = max{d(u, v)|v ∈ V}, where d(u, v) denotes the distance between u and v in H.The first Zagreb index (M 1 ) and second Zagreb index (M 2 ) are the oldest known indices introduced by Gutman and Trinajstić [1] defined as Let , be a simple graph with | | vertices and | | edges.For ∈ , degree of u, denoted by , is number of vertices attached to in the graph.The maximum distance from a vertex to any other vertex in the graph is called eccentricity of the vertex and is denoted by Ɛ i.e., Ɛ max , | ∈ , where , denotes the distance between and in .The first Zagreb index ( ) and second Zagreb index are the oldest known indices introduced by Gutman and Trinajstić [1] defined as In theoretical chemistry, molecular descriptors or topological indices are utilized to co properties of chemical compounds.A topological index is a real number connected with ch structure indicating relationships of chemical configuration with different physical pro chemical reactivity or biological activity, which is utilized to understand properties of ch compounds in theoretical chemistry.Topological indices have been observed to be hel chemical documentation, isomer discrimination, structure-property relations, structure-(SAR) relations and pharmaceutical medication plans.All through the paper, all grap considered to be simple and connected.Let , be a simple graph with | | vertices and | | edges.For ∈ , of u, denoted by , is number of vertices attached to in the graph.The maximum distan a vertex to any other vertex in the graph is called eccentricity of the vertex and is denoted b i.e., Ɛ max , | ∈ , where , denotes the distance between and in .T Zagreb index ( ) and second Zagreb index are the oldest known indices introduced by G and Trinajstić [1] defined as ∈ , .∈ K * m v j = 2, www.mdpi.com/journal/symmetrylege Women University Faisalabad, 38000 Faisalabad, l.com; mber 2016; Published: date e the eccentricity reformulation of the Zagreb indices.centricity index ( ) is defined to be the summation ∑ Ɛ ∈ .The second Zagreb eccentricity f the eccentricities of the adjacent vertices, i.e., graph of a graph by attaching thorns i.e., vertices of er, we will find closed formulation for the first Zagreb tricity index of different well known classes of thorny ; path; star; cycle riptors or topological indices are utilized to configure gical index is a real number connected with chemical cal configuration with different physical properties, hich is utilized to understand properties of chemical logical indices have been observed to be helpful in ation, structure-property relations, structure-activity ation plans.All through the paper, all graphs are | | vertices and | | edges.For ∈ , degree ttached to in the graph.The maximum distance from called eccentricity of the vertex and is denoted by Ɛ denotes the distance between and in .The first are the oldest known indices introduced by Gutman = 3 for j = 1, 2, . . ., m; k = 1, 2, . . ., p j are the eccentricities of the vertices of K * m .Thus, the first Zagreb eccentricity index and the second Zagreb eccentricity index of K * m are given by

1 lege∈.
University Faisalabad, 38000 Faisalabad, Pakistan; nt College University Faisalabad, 38000 Faisalabad, Pakistan; lege Women University Faisalabad, 38000 Faisalabad, l.com; mber 2016; Published: date e the eccentricity reformulation of the Zagreb indices.centricity index ( ) is defined to be the summation ∑ Ɛ The second Zagreb eccentricity f the eccentricities of the adjacent vertices, i.e., graph of a graph by attaching thorns i.e., vertices of er, we will find closed formulation for the first Zagreb tricity index of different well known classes of thorny ; path; star; cycle riptors or topological indices are utilized to configure gical index is a real number connected with chemical cal configuration with different physical properties, hich is utilized to understand properties of chemical logical indices have been observed to be helpful in ation, structure-property relations, structure-activity ation plans.All through the paper, all graphs are | | vertices and | | edges.For ∈ , degree

2 . 3 .Theorem 3 .
The Thorny Star Graph Suppose that we have the star graph S m = K 1,(m−1) of m vertices.Obviously, E 1 (S m ) = 4m − 3 and E 2 (S m ) = 2(m − 1).Let the thorny star graph S * m be obtained by joining p j pendant vertices to every vertex v j , j = 2, 3, . . ., m and p 1 pendant vertices to the central vertex v 1 of S m .The first Zagreb eccentricity index and the second Zagreb eccentricity index of S *

1 ) = 2 ,First
Let , be a simple graph with | | vertices and | | edges.For ∈ , degree of u, denoted by , is number of vertices attached to in the graph.The maximum distance from a vertex to any other vertex in the graph is called eccentricity of the vertex and is denoted by Ɛ i.e., Ɛ max , | ∈ , where , denotes the distance between and in .The first Zagreb index ( ) and second Zagreb index are the oldest known indices introduced by Gutman and Trinajstić [1] defined as Symmetry 2016, 9, 7; doi: 10.3390/sym9010007 www.mdpi.com/journalArticle In theoretical chemistry, molecular descriptors or topological indices are utilized to c properties of chemical compounds.A topological index is a real number connected with c structure indicating relationships of chemical configuration with different physical pr chemical reactivity or biological activity, which is utilized to understand properties of c compounds in theoretical chemistry.Topological indices have been observed to be he chemical documentation, isomer discrimination, structure-property relations, structure (SAR) relations and pharmaceutical medication plans.All through the paper, all gra considered to be simple and connected.Let , be a simple graph with | | vertices and | | edges.For ∈ of u, denoted by , is number of vertices attached to in the graph.The maximum distan a vertex to any other vertex in the graph is called eccentricity of the vertex and is denoted i.e., Ɛ max , | ∈ , where , denotes the distance between and in .Zagreb index ( ) and second Zagreb index are the oldest known indices introduced by and Trinajstić [1] defined as College University Faisalabad, 38000 Faisalabad, Pakistan; (S.M.) hemistry, Government College University Faisalabad, 38000 Faisalabad, Pakistan;

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Accepted: 22 December 2016; Published: date entricity indices are the eccentricity reformulation of the Zagreb indices.The first Zagreb eccentricity index ( ) is defined to be the summation ity of vertices, i.e., ∑ Ɛ The second Zagreb eccentricity ation of product of the eccentricities of the adjacent vertices, i.e., obtain the thorny graph of a graph by attaching thorns i.e., vertices of x of .In this paper, we will find closed formulation for the first Zagreb cond Zagreb eccentricity index of different well known classes of thorny es; complete graph; path; star; cycle try, molecular descriptors or topological indices are utilized to configure pounds.A topological index is a real number connected with chemical ionships of chemical configuration with different physical properties, logical activity, which is utilized to understand properties of chemical l chemistry.Topological indices have been observed to be helpful in isomer discrimination, structure-property relations, structure-activity rmaceutical medication plans.All through the paper, all graphs are d connected.mple graph with | | vertices and | | edges.For ∈ , degree umber of vertices attached to in the graph.The maximum distance from x in the graph is called eccentricity of the vertex and is denoted by Ɛ ∈ , where , denotes the distance between and in .The first nd Zagreb index are the oldest known indices introduced by Gutman s = 4, for j = 2, 3, . . ., m; k = 1, 2, . . ., p j , Article First and Second Zagreb Eccentricity Indices of Thorny Graphs Nazeran Idrees 1, *, Muhammad Jawwad Saif 2 , Asia Rauf 3 and Saba Mustafa 1 Let be a simple graph.The first Zagreb eccentricity index ( Let , be a simple graph with | | vertices and | | edges.For ∈ , degree of u, denoted by , is number of vertices attached to in the graph.The maximum distance from a vertex to any other vertex in the graph is called eccentricity of the vertex and is denoted by Ɛ i.e., Ɛ max , | ∈ , where , denotes the distance between and in .The first Zagreb index ( ) and second Zagreb index are the oldest known indices introduced by Gutman and Trinajstić [1] defined as 1k ) = 3, for k = 1, 2, . . ., p 1 .Thus, the Zagreb eccentricity indices of S * m are

Theorem 4 . 1
the thorny cycle of C m obtained by joining p j thorns v jk to each vertex v j , j = 1, 2, . . ., m of C m .The first Zagreb eccentricity index and the second Zagreb eccentricity index of C * m are given by E Zagreb eccentricity index of C * m is

3, eb Eccentricity Indices of Thorny Graphs d Jawwad Saif 2 , Asia Rauf 3 and Saba Mustafa 1 overnment
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