{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:56:16Z","timestamp":1760230576092,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,2]],"date-time":"2022-08-02T00:00:00Z","timestamp":1659398400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Xinjiang (NSFXJ)","award":["2021D01A65","2019D01B10"],"award-info":[{"award-number":["2021D01A65","2019D01B10"]}]},{"name":"\u201ctian shan ying cai\u201d project for innovative talents of Xinjiang","award":["2021D01A65","2019D01B10"],"award-info":[{"award-number":["2021D01A65","2019D01B10"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, the performance indices on consensus problems for three-layered, multiagent systems are studied from the perspective of algebraic graph theory, where the indices can be used as a measurement of the system performance and refer to the network coherence and algebraic connectivity. Specifically, some operations of two graphs are applied to established the three-layered networks based on k-partite structure, and the mathematical expression of the coherence is derived by the methods of algebraic graph theory. We found that the operations of adding star-shaped copies or fan-graph copies will make the coherence increase by some scalars under the computations of limitation. Then, the indices of the three-layered systems with non-isomorphic topologies but the same number of nodes were compared and simulated; it is found that, when the number of nodes in the counterpart node classes tend to infinity, their difference in coherence are only relevant with the number of peripheral nodes in the sense of limitation.<\/jats:p>","DOI":"10.3390\/sym14081586","type":"journal-article","created":{"date-parts":[[2022,8,3]],"date-time":"2022-08-03T00:15:26Z","timestamp":1659485726000},"page":"1586","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Consensus Indices of Triplex Multiagent Networks Based on Complete k-Partite Graph"],"prefix":"10.3390","volume":"14","author":[{"given":"Da","family":"Huang","sequence":"first","affiliation":[{"name":"Department of Mathematics and Physics, Xinjiang Institute of Engineering, Urumqi 830023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jibin","family":"Yang","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Data Science, Changji University, Changji 831100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xing","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Physics, Xinjiang Institute of Engineering, Urumqi 830023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaolin","family":"Fan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Physics, Xinjiang Institute of Engineering, Urumqi 830023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Kapsoulis, N., Psychas, A., Litke, A., and Varvarigou, T. 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