{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,8]],"date-time":"2026-07-08T21:59:25Z","timestamp":1783547965175,"version":"3.55.0"},"reference-count":9,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2006,10,6]],"date-time":"2006-10-06T00:00:00Z","timestamp":1160092800000},"content-version":"vor","delay-in-days":4602,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1994,3]]},"abstract":"Abstract<\/jats:title>It is well known that the smallest eigenvalue of the adjacency matrix of a connected d<\/jats:italic>\u2010regular graph is at least \u2212 d<\/jats:italic> and is strictly greater than \u2212 d<\/jats:italic> if the graph is not bipartite. More generally, for any connected graph G = (V, E)<\/jats:italic>, consider the matrix Q = D + A<\/jats:italic> where D<\/jats:italic> is the diagonal matrix of degrees in the graph G<\/jats:italic> and A<\/jats:italic> is the adjacency matrix of G<\/jats:italic>. Then Q<\/jats:italic> is positive semidefinite, and the smallest eigenvalue of Q<\/jats:italic> is 0 if and only if G<\/jats:italic> is bipartite. We will study the separation of this eigenvalue from 0 in terms of the following measure of nonbipartiteness of G.<\/jats:italic> For any S \u2286 V<\/jats:italic>, we denote by e<\/jats:italic>min<\/jats:sub>(S<\/jats:italic>) the minimum number of edges that need to be removed from the induced subgraph on S<\/jats:italic> to make it bipartite. Also, we denote by cut(S<\/jats:italic>) the set of edges with one end in S<\/jats:italic> and the other in V<\/jats:italic> \u2212 S<\/jats:italic>. We define the parameter \u03a8 as.<\/jats:p>magnified image<\/jats:alt-text><\/jats:graphic><\/jats:chem-struct><\/jats:chem-struct-wrap><\/jats:p>The parameter \u03a8 is a measure of the nonbipartiteness of the graph G<\/jats:italic>. We will show that the smallest eigenvalue of Q<\/jats:italic> is bounded above and below by functions of \u03a8. For d<\/jats:italic>\u2010regular graphs, this characterizes the separation of the smallest eigenvalue of the adjacency matrix from \u2212d<\/jats:italic>. These results can be easily extended to weighted graphs.<\/jats:p>","DOI":"10.1002\/jgt.3190180210","type":"journal-article","created":{"date-parts":[[2007,6,7]],"date-time":"2007-06-07T17:10:48Z","timestamp":1181236248000},"page":"181-194","source":"Crossref","is-referenced-by-count":110,"title":["A characterization of the smallest eigenvalue of a graph"],"prefix":"10.1002","volume":"18","author":[{"given":"Madhav","family":"Desai","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vasant","family":"Rao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2006,10,6]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"crossref","unstructured":"D.Aldous On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing.Prob. Engin. Inform. Sci.(1987)33\u201346.","DOI":"10.1017\/S0269964800000267"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579166"},{"key":"e_1_2_1_4_2","volume-title":"Spectra of Graphs","author":"Cvetkovic D. M.","year":"1980"},{"key":"e_1_2_1_5_2","unstructured":"M. P.Desai \u201cAn eigenvalue based approach to the finite time behavior of simulated annealing\u201d Ph.D. thesis Department of Electrical and Computer Engineering. University of Illinois at Urbana\u2013Champaign (1991)."},{"key":"e_1_2_1_6_2","doi-asserted-by":"crossref","unstructured":"P.DiaconisandD.Stroock Geometic bounds for eigenvalues of Markov chains. Preprint (1990).","DOI":"10.1214\/aoap\/1177005980"},{"issue":"100","key":"e_1_2_1_7_2","doi-asserted-by":"crossref","first-page":"339","DOI":"10.21136\/CMJ.1975.101329","article-title":"A quanitative extension of the Perron\u2010Frobenius theorem for doubly stochastic matrices","volume":"25","author":"Fiedler M.","year":"1975","journal-title":"Czech. Math. J."},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.2307\/2000925"},{"key":"e_1_2_1_9_2","unstructured":"B.Mohar The Laplacian spectrum of graphs. InternationalX Graph Theory Conference Western Michigan University June (1988)."},{"key":"e_1_2_1_10_2","volume-title":"Nonnegative Matrices and Markov Chains","author":"Senata E.","year":"1980"}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190180210","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190180210","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,24]],"date-time":"2023-10-24T06:58:03Z","timestamp":1698130683000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190180210"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,3]]},"references-count":9,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1994,3]]}},"alternative-id":["10.1002\/jgt.3190180210"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190180210","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994,3]]}}}