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Symmetry 2012, 4(2), 329-335; doi:10.3390/sym4020329

Article

Topological Invariance under Line Graph Transformations

Allen D. Parks

Electromagnetic and Sensor Systems Department, Naval Surface Warfare Center Dahlgren Division, 18444 Frontage Road Suite 327, Dahlgren, VA 22448-5161, USA

Received: 6 April 2012; in revised form: 1 June 2012 / Accepted: 4 June 2012 / Published: 8 June 2012

Download PDF Full-Text [177 KB, uploaded 8 June 2012 14:02 CEST]

Abstract: It is shown that the line graph transformation G ↦ L(G) of a graph G preserves an isomorphic copy of G as the nerve of a finite simplicial complex K which is naturally associated with the Krausz decomposition of L(G). As a consequence, the homology of K is isomorphic to that of G. This homology invariance algebraically confirms several well known graph theoretic properties of line graphs and formally establishes the Euler characteristic of G as a line graph transformation invariant.

Keywords: algebraic graph theory; line graph; Krausz decomposition; homology; graph invariant; Euler characteristic

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