Involve: A Journal of Mathematics

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

Convex and subharmonic functions on graphs

Matthew Burke and Tony Perkins

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513733659

Digital Object Identifier
doi:10.2140/involve.2014.7.227

Mathematical Reviews number (MathSciNet)
MR3133721

Zentralblatt MATH identifier
1282.05059

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

Keywords
convex subharmonic discrete graphs

Citation

Burke, Matthew; Perkins, Tony. Convex and subharmonic functions on graphs. Involve 7 (2014), no. 2, 227--237. doi:10.2140/involve.2014.7.227. https://projecteuclid.org/euclid.involve/1513733659


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References

  • T. B\iy\ikoğlu, J. Leydold, and P. F. Stadler, Laplacian eigenvectors of graphs: Perron–Frobenius and Faber–Krahn type theorems, Lecture Notes in Mathematics 1915, Springer, Berlin, 2007.
  • J. Cáceres, A. Márquez, O. R. Oellermann, and M. L. Puertas, “Rebuilding convex sets in graphs”, Discrete Math. 297:1-3 (2005), 26–37.
  • M. Farber and R. E. Jamison, “Convexity in graphs and hypergraphs”, SIAM J. Algebraic Discrete Methods 7:3 (1986), 433–444.
  • L. F. German, V. P. Soltan, and P. S. Soltan, “Certain properties of $d$-convex sets”, Dokl. Akad. Nauk SSSR 212 (1973), 1276–1279. In Russian; translated in Sov. Math. Dokl. 14 (1973), 1566–1570.
  • C. O. Kiselman, “Regularity of distance transformations in image analysis”, Computer Vision and Image Understanding 64:3 (1996), 390–398.
  • C. O. Kiselman, “Convex functions on discrete sets”, pp. 443–457 in Combinatorial image analysis (Auckland, 2004), edited by R. Klette and J. Žunić, Lecture Notes in Comput. Sci. 3322, Springer, Berlin, 2004.
  • C. O. Kiselman, “Subharmonic functions on discrete structures”, pp. 67–80 in Harmonic analysis, signal processing, and complexity (Fairfax, VA, 2004), edited by I. Sabadini et al., Progr. Math. 238, Birkhäuser, Boston, 2005.
  • P. M. Soardi, Potential theory on infinite networks, Lecture Notes in Mathematics 1590, Springer, Berlin, 1994.
  • P. S. Soltan, “Helly's theorem for $d$-convex sets”, Dokl. Akad. Nauk SSSR 205 (1972), 537–539. In Russian; translated in Sov. Math. Dokl. 13 (1972), 975–978.
  • V. P. Soltan, “$d$-convexity in graphs”, Dokl. Akad. Nauk SSSR 272:3 (1983), 535–537. In Russian; translated in Sov. Math. Dokl. 28 (1983), 419–421.
  • V. P. Soltan, “Metric convexity in graphs”, Studia Univ. Babeş-Bolyai Math. 36:4 (1991), 3–43.
  • V. P. Soltan and P. S. Soltan, “$d$-convex functions”, Dokl. Akad. Nauk SSSR 249:3 (1979), 555–558. In Russian; translated in Sov. Math. Dokl. 20 (1979), 1323–1326.
  • W. Woess, “Random walks on infinite graphs and groups: a survey on selected topics”, Bull. London Math. Soc. 26:1 (1994), 1–60.