1514 KB48 KB200820252019Kubota, ShoTaniguchi, TetsujiYoshino, Kiyotoletnik:17številka:2str. 555-579ISSN:1855-3966COBISSID_HOST:18965593URN:URN:NBN:SI:doc-EXBJK8SJenUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneaHoffman graphHoffmanov grafline graphnajmanjša lastna vrednostpovezavni grafsmallest eigenvalueOn graphs with the smallest eigenvalue at least -1- square root 2, part III|There are many results on graphs with the smallest eigenvalue at least ?$-2$?. In order to study graphs with the eigenvalues at least ?$-1 - \sqrt{2}$?, R. Woo and A. Neumaie Linear Algebra Appl. 226-228, 577-591 (1995) introduced Hoffman graphs and ?$\mathcal{H}$?-line graphs. They proved that a graph with the sufficiently large minimum degree and the smallest eigenvalue at least ?$-1 - \sqrt{2}$? is a slim ?$\{\mathfrak h_2, \mathfrak h_5, \mathfrak h_7, \mathfrak h_9\}$?-line graph. After that, the second author Ars Math. Contemp. 5, No. 2, 243--258 (2012) researched on slim ?$\{\mathfrak h_2, \mathfrak h_5\}$?-line graphs. As an analogue, we reveal the condition under which a strict ?$\{\mathfrak h_1, \mathfrak h_4, \mathfrak h_7\}$?-cover of a slim ?$\{\mathfrak h_7\}$?-line graph is unique, and completely determine the minimal forbidden graphs for the slim ?$\{\mathfrak h_7\}$?-line graphsObstajajo številni rezultati o grafih, katerih najmanjša lastna vrednost je vsaj ?$-2$?. Kot orodje za raziskavo grafov, katerih lastne vrednosti so vsaj ?$-1 - \sqrt{2}$?, sta R. Woo in A. Neumaier vpeljala Hoffmanove grafe in ?$\mathcal{H}$?-povezavne grafe. Dokazala sta, da je graf, ki ima dovolj veliko najmanjšo stopnjo in katerega najmanjša lastna vrednost je vsaj ?$-1 - \sqrt{2}$?, tanek ?$\{\mathfrak h_2, \mathfrak h_5, \mathfrak h_7, \mathfrak h_9\}$?-povezavni graf. Po tem je T. Taniguchi raziskoval tanke ?$\{\mathfrak h_2, \mathfrak h_5\}$?-povezavne grafe. Po analogiji v tem članku razkrivamo pogoje, pod katerimi je strogi ?$\{\mathfrak h_1, \mathfrak h_4, \mathfrak h_7\}$-krov tankega $\{\mathfrak h_7\}$?-povezavnega grafa en sam, in popolnoma določimo minimalne prepovedane grafe za tanke $?\{\mathfrak h_7\}$?-povezavne grafeTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije