First and Second Zagreb Eccentricity Indices 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; [email protected] (S.M.) 2 Department of Applied Chemistry, Government College University Faisalabad, 38000 Faisalabad, Pakistan; [email protected] (M.J.S.); 3 Department of Mathematics, Government College Women University Faisalabad, 38000 Faisalabad, Pakistan; [email protected] (A.R.) * Corresponding author: [email protected]; 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.