{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T22:37:11Z","timestamp":1765233431622,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,11,28]],"date-time":"2022-11-28T00:00:00Z","timestamp":1669593600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Science Foundation of China","doi-asserted-by":"publisher","award":["11561056","2022-ZJ-924"],"award-info":[{"award-number":["11561056","2022-ZJ-924"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Science Foundation of  Qinghai Province","award":["11561056","2022-ZJ-924"],"award-info":[{"award-number":["11561056","2022-ZJ-924"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A singular graph G, defined when its adjacency matrix is singular, has important applications in mathematics, natural sciences and engineering. The chemical importance of singular graphs lies in the fact that if the molecular graph is singular, the nullity (the number of the zero eigenvalue) is greater than 0, then the corresponding chemical compound is highly reactive or unstable. By this reasoning, chemists have a great interest in this problem. Thus, the problem of characterization singular graphs was proposed and raised extensive studies on this challenging problem thereafter. The graph obtained by conglutinating the starting vertices of three paths Ps1, Ps2, Ps3 into a vertex, and three end vertices into a vertex on the cycle Ca1, Ca2, Ca3, respectively, is denoted as \u03b3(a1,a2,a3,s1,s2,s3). Note that \u03b4(a1,a2,a3,s1,s2)=\u03b3(a1,a2,a3,s1,1,s2), \u03b6(a1,a2,a3,s)=\u03b3(a1,a2,a3,1,1,s), \u03c6(a1,a2,a3)=\u03b3(a1,a2,a3,1,1,1). In this paper, we give the necessity and sufficiency that the \u03b3\u2212graph, \u03b4\u2212graph, \u03b6\u2212graph and \u03c6\u2212graph are singular and prove that the probability that a randomly given \u03b3\u2212graph, \u03b4\u2212graph, \u03b6\u2212graph or \u03c6\u2212graph being singular is equal to 325512,165256,4364, 2132, respectively. From our main results, we can conclude that such a \u03b3\u2212graph(\u03b4\u2212graph, \u03b6\u2212graph, \u03c6\u2212graph) is singular if at least one cycle is a multiple of 4 in length, and surprisingly, the theoretical probability of these graphs being singular is more than half. This result promotes the understanding of a singular graph and may be promising to propel the solutions to relevant application problems.<\/jats:p>","DOI":"10.3390\/sym14122507","type":"journal-article","created":{"date-parts":[[2022,11,28]],"date-time":"2022-11-28T04:01:33Z","timestamp":1669608093000},"page":"2507","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["The Singularity of Four Kinds of Tricyclic Graphs"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5503-4668","authenticated-orcid":false,"given":"Haicheng","family":"Ma","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Qinghai Minzu University, Xining 810007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shang","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Qinghai Minzu University, Xining 810007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bin","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Qinghai Minzu University, Xining 810007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3827","DOI":"10.1021\/cr0004456","article-title":"From molecular connectivity indices to semiempirical connectivity terms: Recent trends in graph theoretical descriptors","volume":"100","author":"Pogliani","year":"2000","journal-title":"Chem. 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