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Exponential generalised network descriptors

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  • In communication networks theory the concepts of networkness and network surplus have recently been defined. Together with transmission and betweenness centrality, they were based on the assumption of equal communication between vertices. Generalised versions of these four descriptors were presented, taking into account that communication between vertices $ u $ and $ v $ is decreasing as the distance between them is increasing. Therefore, we weight the quantity of communication by $ \lambda^{d(u,v)} $ where $ \lambda \in \left\langle0,1 \right\rangle $. Extremal values of these descriptors are analysed.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

    Citation:

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  • Figure 1.  A broom that minimizes $mt_{\lambda }^{e}(G)$

    Table 1.  Extremal values of exponential generalised network descriptors

    Descriptor $\lambda \in \left\langle 0,1\right\rangle $
    Lower bound Upper bound
    $mt_{\lambda }^{e}$ broom (starting vertex) complete graph *
    $A_n$ $ (n-1)\lambda $
    $Mt_{\lambda }^{e}$ open problem broom (starting vertex)
    $B_n$
    $mc_{\lambda }^{e}$ path (end vertices) complete graph *
    $\frac{\lambda^D-\lambda}{\lambda -1}$ $(n-1)\lambda $
    $Mc_{\lambda }^{e}$ open problem star (center)
    $(n-1)\left[ \lambda +\frac{1}{2}(n-2)\lambda ^{2}\right] $
    $mN_{\lambda }^{e}$ broom (starting vertex) vertex-transitive graph
    $C_n$ $1$
    $MN_{\lambda }^{e}$ vertex-transitive graph star (center)
    $1$ $\frac{1}{2}(n-2)\lambda +1$
    $m\nu _{\lambda }^{e}$ broom (starting vertex) vertex-transitive graph
    $D_n$ $0$
    $M\nu _{\lambda }^{e}$ vertex-transitive graph star (center)
    $0$ $\frac{1}{2}(n-1)(n-2)\lambda ^{2}$
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