Exponential generalised network descriptors

Suzana Antunović; Tonči Kokan; Tanja Vojković; Damir Vukičević

doi:10.3934/amc.2019026

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2019, Volume 13, Issue 3: 405-420. Doi: 10.3934/amc.2019026

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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
^* Corresponding author

Received: May 2018

Revised: February 2019

Published online: August 2019

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Abstract

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.

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

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

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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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References

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