{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T02:49:02Z","timestamp":1773456542007,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2017,1,6]],"date-time":"2017-01-06T00:00:00Z","timestamp":1483660800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The Zagreb eccentricity indices are the eccentricity reformulation of the Zagreb indices. Let    H    be a simple graph. The first Zagreb eccentricity index (     E 1   ( H )     ) is defined to be the summation of squares of the eccentricity of vertices, i.e.,      E 1   ( H )  =  \u2211  u \u2208 V  ( H )     \u0190 H 2   ( 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 )  =  \u2211  u v \u2208 E  ( H )     \u0190 H   ( u )   \u0190 H   ( v )     . We obtain the thorny graph of a graph    H    by attaching thorns i.e., vertices of degree one to every vertex of    H   . 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.<\/jats:p>","DOI":"10.3390\/sym9010007","type":"journal-article","created":{"date-parts":[[2017,1,6]],"date-time":"2017-01-06T10:08:12Z","timestamp":1483697292000},"page":"7","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["First and Second Zagreb Eccentricity Indices of Thorny Graphs"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2578-9816","authenticated-orcid":false,"given":"Nazeran","family":"Idrees","sequence":"first","affiliation":[{"name":"Department of Mathematics, Government College University Faisalabad, 38000 Faisalabad, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9589-876X","authenticated-orcid":false,"given":"Muhammad","family":"Saif","sequence":"additional","affiliation":[{"name":"Department of Applied Chemistry, Government College University Faisalabad, 38000 Faisalabad, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Asia","family":"Rauf","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College Women University Faisalabad, 38000 Faisalabad, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Saba","family":"Mustafa","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University Faisalabad, 38000 Faisalabad, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,1,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1016\/0009-2614(72)85099-1","article-title":"Graph theory and molecular orbitals. 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