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  • The Wiener index in iterated line graphs
    Knor, Martin ; Potočnik, Primož, 1971- ; Škrekovski, Riste
    For a graph ▫$G$▫, denote by ▫$L^i(G)$▫ its ▫$i$▫-iterated line graph and denote by ▫$W(G)$▫ its Wiener index. We prove that the function ▫$W(L^i(G))$▫ is convex invariable ▫$i$▫. Moreover, this ... function is strictly convex if ▫$G$▫ is different from a path, a claw ▫$K_{1,3}$▫ and a cycle. As an application we prove that ▫$W(L^i(T)) \ne W(T)$▫ for every ▫$i\ge 3$▫ if ▫$T$▫ is a tree in which no leaf is adjacent to a vertex of degree ▫$2$▫, ▫$T\ne K_1$▫ and ▫$T\ne K_2$▫. This is the first in a series of papers which resolve the question, whether there exists a tree ▫$T$▫ for which ▫$W(L^k(T)) = W(L(T))$▫ for some ▫$k \ge 3$▫, which was posed in [A.A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: Theory and applications, Acta Appl. Math. 66 (2001), 211-249].
    Vir: Discrete applied mathematics. - ISSN 0166-218X (Vol. 160, iss. 15, 2012, str. 2234-2345)
    Vrsta gradiva - članek, sestavni del
    Leto - 2012
    Jezik - angleški
    COBISS.SI-ID - 16409945

    Povezava(-e):

    http://dx.doi.org/10.1016/j.dam.2012.04.021

vir: Discrete applied mathematics. - ISSN 0166-218X (Vol. 160, iss. 15, 2012, str. 2234-2345)

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