{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:27:19Z","timestamp":1760239639058,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,12,8]],"date-time":"2020-12-08T00:00:00Z","timestamp":1607385600000},"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>If G is a graph and P is a partition of V(G), then the partition distance of G is the sum of the distances between all pairs of vertices that lie in the same part of P. A colored distance is the dual concept of the partition distance. These notions are motivated by a problem in the facility location network and applied to several well-known distance-based graph invariants. In this paper, we apply an extended cut method to induce the partition and color distances to some subsets of vertices which are not necessary a partition of V(G). Then, we define a two-dimensional weighted graph and an operator to prove that the induced partition and colored distances of a graph can be obtained from the weighted Wiener index of a two-dimensional weighted quotient graph induced by the transitive closure of the Djokovi\u0107\u2013Winkler relation as well as by any partition that is coarser. Finally, we utilize our main results to find some upper bounds for the modified Wiener index and the number of orbits of partial cube graphs under the action of automorphism group of graphs.<\/jats:p>","DOI":"10.3390\/sym12122027","type":"journal-article","created":{"date-parts":[[2020,12,10]],"date-time":"2020-12-10T22:15:36Z","timestamp":1607638536000},"page":"2027","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Partition and Colored Distances in Graphs Induced to Subsets of Vertices and Some of Its Applications"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1754-6694","authenticated-orcid":false,"given":"Mohammad Javad","family":"Nadjafi-Arani","sequence":"first","affiliation":[{"name":"Faculty of Science, Mahallat Institute of Higher Education, Mahallat 3781151958, Iran"},{"name":"Steyr School of Management, University of Applied Sciences of Upper Austria, 4400 Steyr, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mahsa","family":"Mirzargar","sequence":"additional","affiliation":[{"name":"Faculty of Science, Mahallat Institute of Higher Education, Mahallat 3781151958, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Frank","family":"Emmert-Streib","sequence":"additional","affiliation":[{"name":"Predictive Society and Data Analytics Lab., Faculty of Information Technology and Communication Sciences, Tampere University, 33100 Tampere, Finland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Matthias","family":"Dehmer","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Swiss Distance University of Applied Sciences, 3900 Brig, Switzerland"},{"name":"College of Artificial Intelligence, Nankai University, Tianjin 300350, China"},{"name":"School of Sciences, Xi\u2019an Technological University, Xi\u2019an 710021, China"},{"name":"Department of Biomedical Computer Science and Mechatronics, UMIT, A-6060 Hall in Tyrol, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/jgt.1020","article-title":"Average distance in colored graphs","volume":"38","author":"Dankelmann","year":"2001","journal-title":"J. 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