On the ▫$A_\alpha$▫-spectral radius of connected graphs
Alhevaz, Abdollah ...
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 sosednosti.
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