| [1] |
G. Fan.
Subgraph coverings and edge switchings.
Journal of Combinatorial Theory, 84(1):54-83, January 2002.
Series B. BibTeX entry, PDF
By using the so-defined circuit/path transformations together with an edge-switching method, the following conjectures are proved in this paper. (i) The edges of a connected graph on n vertices can be covered by at most paths, which was conjectured by Chung. (ii) The edges of a 2-connected graph on n vertices can be covered by at most circuits, which was conjectured by Bondy. An immediate consequence of (ii) is a theorem of Pyber that the edges of a graph on n vertices can be covered by at most n-1 edges and circuits, which was conjectured by Erdös, Goodman, and Pósa.
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| [2] |
Christophe Texier.
Scattering theory on graphs: Ii. the friedel sum rule.
Journal of Physics A: Mathematical and General,
35(15):3389-3407, 2002. BibTeX entry, PDF
We consider the Friedel sum rule (FSR) in the context of the scattering theory for the Schrödinger operator -D x^2 + V ( x ) on graphs made of one-dimensional wires connected to external leads. We generalize the Smith formula for graphs. We give several examples of graphs where the state counting method given by the FSR does not work. The reason for the failure of the FSR to count the states is the existence of states localized in the graph and not coupled to the leads, which occurs if the spectrum is degenerate and the number of leads too small.
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