Involve: A Journal of Mathematics

  • Involve
  • Volume 7, Number 2 (2014), 227-237.

Convex and subharmonic functions on graphs

Matthew Burke and Tony Perkins

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We explore the relationship between convex and subharmonic functions on discrete sets. Our principal concern is to determine the setting in which a convex function is necessarily subharmonic. We initially consider the primary notions of convexity on graphs and show that more structure is needed to establish the desired result. To that end, we consider a notion of convexity defined on lattice-like graphs generated by normed abelian groups. For this class of graphs, we are able to prove that all convex functions are subharmonic.

Article information

Involve, Volume 7, Number 2 (2014), 227-237.

Received: 1 April 2013
Revised: 21 June 2013
Accepted: 5 July 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A51: Convexity, generalizations
Secondary: 31C20: Discrete potential theory and numerical methods

convex subharmonic discrete graphs


Burke, Matthew; Perkins, Tony. Convex and subharmonic functions on graphs. Involve 7 (2014), no. 2, 227--237. doi:10.2140/involve.2014.7.227.

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