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Extremal ▫$(n,m)$▫-graphs with respect to distance-degree-based topological indices
Xu, Kexiang ...
V kemijski teoriji grafov so topološki indeksi, ki so zasnovani na razdalji in stopnji, izrazi oblike ▫$\sum_{u \ne v}F({\rm deg}(u), {\rm deg}(v), d(u,v))$▫, kjer je ▫$F$▫ funkcija, ▫${\rm deg}(u)$▫ ...stopnja od ▫$u$▫, in ▫$d(u,v)$▫ razdalja med ▫$u$▫ in ▫$v$▫. Če za ▫$F$▫ izberemo ▫$({\rm deg}(u) + {\rm deg}(v))d(u,v)$▫, ▫${\rm deg}(u){\rm deg}(v)d(u,v)$▫, ▫$({\rm deg}(u) + {\rm deg}(v))d(u,v)^{-1}$▫, in ▫${\rm deg}(u){\rm deg}(v)d(u,v)^{-1}$▫, po vrsti dobimo indeks stopnje in razdalje ▫${\rm DD}$▫, Gutmanov indeks ▫${\rm Gut}$▫, aditivno utežen Hararyjev indeks ▫$H_A$▫, in multiplikativno utežen Hararyjev indeks ▫$H_M$▫. Naj bo ▫${\cal{G}}_{n,m}$▫ množica povezanih grafov z ▫$n$▫ vozlišči in ▫$m$▫ povezavami. Dokazano je, da če je ▫$G\in {\cal{G}}_{n,m}$▫, kjer je ▫$4 \le n \leq m \leq 2n-4$▫, potem velja ▫$H_A(G) \leq (m(m+5) + 2(n-1)(n-3))/2$▫ in ▫${\rm DD}(G) \geq (4m-n)(n-1)-(m-n+1)(m-n+6)$▫. V obeh primerih so karakterizirani ekstremni grafi (in so enaki). Najdeni so tudi ekstremni grafi iz ▫${\cal{G}}_{n,m}$▫ z ▫$m = n + {k\choose 2}-k$▫, ▫$2 \le k \le n-1$▫, ki maksimizirajo multiplikativni uteženi Hararyjev indeks in ki minimizirajo Gutmanov index.
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