{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,13]],"date-time":"2025-09-13T16:13:25Z","timestamp":1757780005879,"version":"3.37.3"},"reference-count":14,"publisher":"EDP Sciences","issue":"6","license":[{"start":{"date-parts":[[2024,12,6]],"date-time":"2024-12-06T00:00:00Z","timestamp":1733443200000},"content-version":"vor","delay-in-days":35,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"have no funding"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2024,9,5]]},"published-print":{"date-parts":[[2024,11]]},"abstract":"<jats:p>For a graph <jats:italic>G<\/jats:italic>, the <jats:italic>k<\/jats:italic>-distance degree index is<\/jats:p>\n<jats:p>\n<jats:italic>N<jats:sub>\u03ba<\/jats:sub>(G)<\/jats:italic> = \u2211<jats:sub><jats:italic>\u03ba<\/jats:italic>=1<\/jats:sub><jats:sup><jats:italic>diam(G)<\/jats:italic><\/jats:sup>\u200a( \u2211<jats:sub><jats:italic>w\u2208V(G)<\/jats:italic><\/jats:sub>\u200a <jats:italic>d<jats:sub>\u03ba<\/jats:sub>(w)<\/jats:italic> ) <jats:italic>\u03ba<\/jats:italic>.\n<\/jats:p>\n<jats:p>In this article, we calculate the accurate formula of the expected value for <jats:italic>k<\/jats:italic>-distance degree index in a random arranged polygonal string with <jats:italic>m<\/jats:italic>-arranged polygons. Finally, we determine the average value of this index in the collection of all those arranged polygonal strings.<\/jats:p>","DOI":"10.1051\/ro\/2024175","type":"journal-article","created":{"date-parts":[[2024,9,6]],"date-time":"2024-09-06T18:58:41Z","timestamp":1725649121000},"page":"5159-5166","source":"Crossref","is-referenced-by-count":1,"title":["Expected value for the <i>k<\/i>-distance degree index of a graph"],"prefix":"10.1051","volume":"58","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3894-9214","authenticated-orcid":false,"given":"Hamideh","family":"Aram","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"250","published-online":{"date-parts":[[2024,12,6]]},"reference":[{"key":"R1","first-page":"81","volume":"6","author":"Ali","year":"2021","journal-title":"Commun. Comb. Optim."},{"key":"R2","first-page":"87","volume":"2","author":"Aram","year":"2017","journal-title":"Commun. Comb. 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