{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T00:45:49Z","timestamp":1768437949888,"version":"3.49.0"},"reference-count":21,"publisher":"Frontiers Media SA","license":[{"start":{"date-parts":[[2022,10,14]],"date-time":"2022-10-14T00:00:00Z","timestamp":1665705600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["frontiersin.org"],"crossmark-restriction":true},"short-container-title":["Front. Comput. Neurosci."],"abstract":"<jats:p>Let <jats:italic>G<\/jats:italic> be a graph, and the number of components of <jats:italic>G<\/jats:italic> is denoted by <jats:italic>c<\/jats:italic>(<jats:italic>G<\/jats:italic>). Let <jats:italic>t<\/jats:italic> be a positive real number. A connected graph <jats:italic>G<\/jats:italic> is <jats:italic>t<\/jats:italic>-tough if <jats:italic>tc<\/jats:italic>(<jats:italic>G<\/jats:italic> \u2212 <jats:italic>S<\/jats:italic>) \u2264 |<jats:italic>S<\/jats:italic>| for every vertex cut <jats:italic>S<\/jats:italic> of <jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>). The <jats:italic>toughness<\/jats:italic> of <jats:italic>G<\/jats:italic> is the largest value of <jats:italic>t<\/jats:italic> for which <jats:italic>G<\/jats:italic> is <jats:italic>t<\/jats:italic>-tough, denoted by \u03c4(<jats:italic>G<\/jats:italic>). We call a graph <jats:italic>G<\/jats:italic> Hamiltonian if it has a cycle that contains all vertices of <jats:italic>G<\/jats:italic>. Chv\u00e1tal and other scholars investigate the relationship between toughness conditions and the existence of cyclic structures. In this paper, we establish some sufficient conditions that a graph with toughness is Hamiltonian based on the number of edges, spectral radius, and signless Laplacian spectral radius of the graph.<\/jats:p><jats:p><jats:bold>MR subject classifications:<\/jats:bold> 05C50, 15A18.<\/jats:p>","DOI":"10.3389\/fncom.2022.1019039","type":"journal-article","created":{"date-parts":[[2022,10,14]],"date-time":"2022-10-14T06:07:46Z","timestamp":1665727666000},"update-policy":"https:\/\/doi.org\/10.3389\/crossmark-policy","source":"Crossref","is-referenced-by-count":4,"title":["Some sufficient conditions on hamilton graphs with toughness"],"prefix":"10.3389","volume":"16","author":[{"given":"Gaixiang","family":"Cai","sequence":"first","affiliation":[]},{"given":"Tao","family":"Yu","sequence":"additional","affiliation":[]},{"given":"Huan","family":"Xu","sequence":"additional","affiliation":[]},{"given":"Guidong","family":"Yu","sequence":"additional","affiliation":[]}],"member":"1965","published-online":{"date-parts":[[2022,10,14]]},"reference":[{"key":"B1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0012-365X(93)E0204-H","article-title":"Long cycles in graphs with prescribed toughness and minimum degree","volume":"141","author":"Bauer","year":"","journal-title":"Discrete Math."},{"key":"B2","unstructured":"\u201cSome recent results on long cycles in tough graphs,\u201d\n            BauerD.\n            SchmeichelE.\n            VeldmanH. 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