American Institute of Mathematical Sciences

August  2019, 13(3): 405-420. doi: 10.3934/amc.2019026

Exponential generalised network descriptors

 1 Faculty of Civil Engineering, Architecture and Geodesy, Matice hrvatske 15, University of Split, Croatia 2 Faculty of Science, Bijenička cesta 30, University of Zagreb, Croatia 3 Faculty of Science, Rudera Boškovića 33, University of Split, Croatia

* Corresponding author

Received  May 2018 Revised  February 2019 Published  April 2019

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.

Citation: Suzana Antunović, Tonči Kokan, Tanja Vojković, Damir Vukičević. Exponential generalised network descriptors. Advances in Mathematics of Communications, 2019, 13 (3) : 405-420. doi: 10.3934/amc.2019026
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References:
A broom that minimizes $mt_{\lambda }^{e}(G)$
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) $0$ $\frac{1}{2}(n-1)(n-2)\lambda ^{2}$
 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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