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KYUNGPOOK Math. J. 2019; 59(4): 797-820

Published online December 23, 2019

Copyright © Kyungpook Mathematical Journal.

Cross-index of a Graph

Akio Kawauchi∗, Ayaka Shimizu and Yoshiro Yaguchi

Osaka City University Advanced Mathematical Institute, Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
e-mail : kawauchi@sci.osaka-cu.ac.jp
Department of Mathematics, National Institute of Technology, Gunma College, 580 Toriba-cho, Maebashi-shi, Gunma 371-8530, Japan
e-mail : shimizu@nat.gunma-ct.ac.jp and yaguchi-y@nat.gunma-ct.ac.jp

Received: January 14, 2018; Revised: February 11, 2019; Accepted: February 13, 2019

Abstract

For every tree T, we introduce a topological invariant, called the T-cross-index, for connected graphs. The T-cross-index of a graph is a non-negative integer or infinity according to whether T is a tree basis of the graph or not. It is shown how this cross-index is independent of the other topological invariants of connected graphs, such as the Euler characteristic, the crossing number and the genus.

Keywords: based graph, crossing number, genus, nullity, cross-index. The fi,rst author was supported by JSPS KAKENHI Grant Number 24244005.