Browsing by Author "Cinkir, Zubeyir"

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Admissible invariants of genus 3 curves
(Springer New York LLC, 2015) Cinkir, Zubeyir; AGÜ, Mühendislik Fakültesi, Endüstri Mühendisliği Bölümü; Çinkir, Zübeyir
Several invariants of polarized metrized graphs and their applications in Arithmetic Geometry are studied recently. In this paper, we explicitly calculated these admissible invariants for all curves of genus 3. We find the sharp lower bound for the invariants φ, λ and ε for all polarized metrized graphs of genus 3. This improves the lower bound given for Effective Bogomolov Conjecture for such curves.
Contraction Formulas for the Kirchhoff and Wiener Indices
(UNIV KRAGUJEVAC, FAC SCIENCEPO BOX 60, RADOJA DOMANOVICA 12, KRAGUJEVAC 34000, SERBIA, 2016) Cinkir, Zubeyir; AGÜ, Mühendislik Fakültesi, Endüstri Mühendisliği Bölümü; Cinkir, Zubeyir
We relate the Kirchhoff index with some other metrized graph invariants. We establish several contraction formulas for the Kirchhoff index. We use these contraction formulas and certain edge densities to give new upper and lower bounds to the Kirchhoff index for any connected graph. As an another application of our contraction formulas when the graph is a tree, we derive new formulas as well as previously known formulas for the Wiener index with new proofs.
Effective resistances and Kirchhoff index of ladder graphs
(SPRINGERONE NEW YORK PLAZA, SUITE 4600 , NEW YORK, NY 10004, UNITED STATES, 2016) Cinkir, Zubeyir; AGÜ, Mühendislik Fakültesi, Endüstri Mühendisliği Bölümü; Cinkir, Zubeyir
We explicitly compute the effective resistances between any two vertices of a ladder graph by using circuit reductions. Using our findings, we obtain explicit formulas for Kirchhoff index of a ladder graph. Comparing our formula for Kirchhoff index and previous results in the literature, we obtain an explicit sum formula involving trigonometric functions. We also expressed our formulas in terms of certain generalized Fibonacci numbers that are the values of the Chebyshev polynomials of the second kind at 2.
Families of Metrized Graphs with Small Tau Constants
(SPRINGER BASEL AGPICASSOPLATZ 4, BASEL 4052, SWITZERLAND, 2016) Cinkir, Zubeyir; AGÜ, Mühendislik Fakültesi, Endüstri Mühendisliği Bölümü; Cinkir, Zubeyir
Baker and Rumely's tau lower bound conjecture claims that if the tau constant of a metrized graph is divided by its total length, this ratio must be bounded below by a positive constant for all metrized graphs. We construct several families of metrized graphs having small tau constants. In addition to numerical computations, we prove that the tau constants of the metrized graphs in one of these families, the hexagonal nets around a torus, asymptotically approach to 108 which is our conjectural lower bound.