453 KB52 KB163 KB5 KB200820252023Alhevaz, AbdollahBaghipur, MaryamDas, Kinkar ChandraGanie, Hilal Ahmadštevilka:1letnik:23P1.06 (16 str.)DOI:10.26493/1855-3974.2697.43aCOBISSID_HOST:149106691ISSN:1855-3966URN:URN:NBN:SI:doc-UB7SW6USenUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneaadjacency matrixclique numberdegree sequencegeneralized adjacency matrixindependence numbermatrika sosednostinepredznačena Laplaceova matrikaposplošena matrika sosednostisignless Laplacian matrixspectral radiusspektralni polmerštevilo klikštevilo neodvisnostizaporedje stopenjOn the Asubalpha-spectral radius of connected graphs|For a simple graph ?$G$?, the generalized adjacency matrix ?$A_\alpha(G)$? is defined as ?$A_\alpha(G) = \alpha D(G) + (1 - \alpha)A(G)$?, ?$\alpha \in 0, 1$?, where ?$A(G)$? is the adjacency matrix and ?$D(G)$? is the diagonal matrix of the vertex degrees. It is clear that ?$A_0(G) = A(G)$ and $2A_{\frac{1}{2}}(G) = Q(G)$? implying that the matrix ?$A_\alpha(G)$? is a generalization of the adjacency matrix and the signless Laplacian matrix. In this paper, we obtain some new upper and lower bounds for the generalized adjacency spectral radius ?$\lambda (A_\alpha(G))$?, in terms of vertex degrees, average vertex 2-degrees, the order, the size, etc. The extremal graphs attaining these bounds are characterized. We will show that our bounds are better than some of the already known bounds for some classes of graphs. We derive a general upper bound for ?$\lambda (A_\alpha(G))$?, in terms of vertex degrees and positive real numbers ?$b_i$?. As application, we obtain some new upper bounds for ?$\lambda (A_\alpha(G))$?. Further, we obtain some relations between clique number ?$\omega(G)$?, independence number ?$\gamma(G)$? and the generalized adjacency eigenvalues of a graph ?$G$?Naj bo ?$G$? enostaven graf; potem je posplošena matrika sosednosti ?$A_\alpha(G)$? definirana kot ?$A_\alpha(G) = \alpha D(G) + (1 - \alpha)A(G)$?, ?$\alpha \in 0, 1$?, kjer je ?$A(G)$? matrika sosednosti in ?$D(G)$? diagonalna matrika stopenj vozlišč. Jasno je, da ?$A_0(G) = A(G)$? in ?$2A_{\frac{1}{2}}(G) = Q(G)$?, kar pomeni, da je matrika ?$A_\alpha(G)$? posplošitev matrike sosednosti in nepredznačene Laplaceove matrike. V članku izpeljemo nekaj novih zgornjih in spodnjih mej za spektralni polmer ?$\lambda (A_\alpha(G))$? posplošene matrike sosednosti glede na stopnje vozlišč, povprečne 2-stopnje vozlišč, red, velikost, itd. Karakteriziramo ekstremne grafe, ki te meje dosegajo. Pokažemo, da so naše meje boljše od nekaterih že znanih mej za nekatere razrede grafov. Izpeljemo splošno zgornjo mejo za ?$\lambda (A_\alpha(G))$? glede na stopnje vozlišč in pozitivna realna števila ?$b_i$?. Z uporabo teh rezultatov dobimo za ?$\lambda (A_\alpha(G))$? nekaj novih zgornjih mej. Izpeljemo tudi nekaj relacij med številom klik ?$\omega(G)$?, številom neodvisnosti ?$\gamma(G)$? in lastnimi vrednostmi grafa ?$G$? za pripadajočo posplošeno matriko sosednostiTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije