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Báo cáo toán học: " The tau constant of a metrized graph and its behavior under graph operations"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: The tau constant of a metrized graph and its behavior under graph operations. | The tau constant of a metrized graph and its behavior under graph operations Zubeyir Cinkir Department of Elementary Mathematics Teaching Zirve University Gaziantep TURKEY zubeyir.cinkir@zirve.edu.tr Submitted Sep 13 2010 Accepted Mar 27 2011 Published Apr 7 2011 Mathematics Subject Classification 05C99 94C99 05C76 Abstract This paper concerns the tau constant which is an important invariant of a metrized graph and which has applications to arithmetic properties of algebraic curves. We give several formulas for the tau constant and show how it changes under graph operations including deletion of an edge contraction of an edge and union of graphs along one or two points. We show how the tau constant changes when edges of a graph are replaced by arbitrary graphs. We prove Baker and Rumely s lower bound conjecture on the tau constant for several classes of metrized graphs. 1 Introduction A metrized graph r is a finite compact topological graph equipped with a distance function on their edges. In this paper we give foundational results on the tau constant T r a positive real-valued number. We systematically study T r and develop a calculus for its computations. Our results in this article and in 3 4 5 6 and 7 are intended to show that T r should be considered a fundamental invariant of r. Our results extend to weighted graphs. We show that many of the intricate calculations concerning metrized graphs can be obtained in much simpler way by viewing the graph as an electrical circuit and then performing suitable circuit reductions. I would like to thank Dr. Robert Rumely for his guidance. His continued support and encouragement made this work possible. I would like to thank Dr. Matthew Baker for always being available for useful discussions during and before the preparation of this paper. Their suggestions and work were inspiring to me. I also would like to thank to anonymous referee for very helpful and detailed feedback on earlier version of this paper. THE ELECTRONIC .