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

Topological Invariance under Line Graph Transformations

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 / Revised: 1 June 2012 / Accepted: 4 June 2012 / Published: 8 June 2012
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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 algebraic graph theory; line graph; Krausz decomposition; homology; graph invariant; Euler characteristic
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Parks, A.D. Topological Invariance under Line Graph Transformations. Symmetry 2012, 4, 329-335.

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