arXiv subjects
Publications and source records attributed to Zubeyir Cinkir.
For any given integer $p$, we show that primality of $p$ is equivalent to $\lceil \frac{1}{2}\log_{2}p \rceil$ congruences of binomial coefficients modulo $p$. We also give extensions of this result to several other congruences that are similar in nature.
We compute the total number of spanning trees for the generalized cone of the complete graph $K_n$ and a number of families of some modified bipartite graphs $K_{m,n}$. In particular, we obtain a new method of finding the number of spanning trees of $K_n$ and $K_{m,n}$. Our method relies on the vertex deletion formula for the number of spanning trees.
We give identities for the voltage and resistance functions on a metrized graph to show how these functions behave under any edge deletion/contraction and the identification of any two vertices. This leads to explicit versions of Rayleigh's Principles on a resistive electrical network. We also establish Euler's Identities for the resistance and the voltage functions on an electrical network. One can use these results to study various invariants of metrized graphs and electrical networks. As a specific application, we obtain various identities for the total number of spanning trees of a graph. For example, we show how the total number of spanning trees changes under graph operations such as, contraction of an edge, deletion of an edge, deletion of a vertex, the join of arbitrary two or three vertices.
Wolstenholme's type summations involve certain powers of all residues $k$ modulo some prime number $p$. We first consider the sums of double or triple products of certain powers of all residues, e.g., the sums of the terms $(a+k)^m(b+k)^n$ or $(a+k)^m(b+k)^n(c+k)^s$ as $k$ ranges over all residues modulo $p$. We consider the sums of double or triple ratios of such terms. We showed that each of such sums is congruent to some simpler expression involving certain binomial coefficients. We also generalize these results to the sums of products or ratios of arbitrary $n$ terms: $(a_1+k)^{m_1}$, ..., $(a_n+k)^{m_n}$. We relate such summations to the sum of certain coefficients of polynomials of type $(a_1-a_n+x)^{m_1} \cdots (a_{n-1}-a_n+x)^{m_{n-1}}$.
We give elementary proofs of some congruence criteria to compute binomial coefficients in modulo a prime. These criteria are analogues to the symmetry property of binomial coefficients. We give extended version of Lucas Theorem by using those criteria. We give applications of these criteria by describing a method to derive identities and congruences involving sums of binomial coefficients.
We explicitly compute the effective resistances between any two vertices of a prism graph by using circuit reductions and our earlier findings on a ladder graph. As an application, we derived a closed form formula for the Kirchhoff index of a prism graph. We show as a byproduct that an explicit sum formula involving trigonometric functions hold by comparing our formula for the Kirchhoff index and previously known results in the literature. We also expressed our formulas in terms of certain generalized Fibonacci numbers.
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 and admissible invariants of a ladder graph considering it as a model of a metrized graph. Comparing our formula for Kirchhoff index and previous results in 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 $2$.
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.
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 $\frac{1}{108}$ which is our conjectural lower bound.
We establish several contraction formulas for Kirchhoff index. We relate Kirchhoff index with some other metrized graph invariants. By applying our contraction formulas successively when the graph is a tree, we derive new formulas for Wiener index and obtain some previously known Wiener index formulas with new proofs.
Arakelov-Green functions defined on metrized graphs have important role in relating arithmetical problems on algebraic curves into graph theoretical problems. In this paper, we clarify the combinatorial interpretation of certain Arakelov-Green functions by using electric circuit theory. The formulas we gave clearly show that such functions are piece-wisely defined, and each piece is a linear or quadratic function on each pair of edges of metrized graphs. These formulas lead to an algorithm for explicit computation of Arakelov-Green functions.
In recent years, a number of fast algorithms for computing the determinant of a Toeplitz matrix were developed. The fastest algorithm we know so far is of order $k^2\log{n}+k^3$, where $n$ is the number of rows of the Toeplitz matrix and $k$ is the bandwidth size. This is possible because such a determinant can be expressed as the determinant of certain parts of $n$-th power of a related $k \times k$ companion matrix. In this paper, we give a new elementary proof of this fact, and provide various examples. We give symbolic formulas for the determinants of Toeplitz matrices in terms of the eigenvalues of the corresponding companion matrices when $k$ is small.
Several invariants of polarized metrized graphs and their applications in Arithmetic Geometry are studied recently. In this paper, we give fast algorithms to compute these invariants by expressing them in terms of the discrete Laplacian matrix and its pseudo inverse. Algorithms we give can be used for both symbolic and numerical computations. We present various examples to illustrate the implementation of these algorithms.
We express the tau constant of a metrized graph in terms of the discrete Laplacian matrix and its pseudo inverse.
In this paper, we give a formula for the area of the triangle formed by the vertices that live on a given polynomial, and we generalize this formula to the volumes of $n$-simplices with vertices on a polynomial space curve. To prove these results, we use induction arguments and a well known identity for complete symmetric polynomials.
Foster's network theorems and their extensions to higher orders involve resistance values and conductances. We establish identities concerning voltage values and conductances. Our identities are analogous to the extended Foster's identities.
We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang's Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures were previously known to be true only for curves of genus at most 4 and a few other special cases. We also either verify or improve the previous results. We relate the invariants involved in Zhang's Conjecture to the tau constant of metrized graphs. Then we use and extend our previous results on the tau constant. By proving another Conjecture of Zhang, we obtain a new proof of the slope inequality for Faltings heights on moduli space of curves.
The tau constant is an important invariant of a metrized graph, and it has applications in arithmetic properties of curves. We show how the tau constant of a metrized graph changes under successive edge contractions and deletions. We discover identities which we call "contraction", "deletion", and "contraction-deletion" identities on a metrized graph. By establishing a lower bound for the tau constant in terms of the edge connectivity, we prove that Baker and Rumely's lower bound conjecture on the tau constant holds for metrized graphs with edge connectivity 5 or more. We show that proving this conjecture for 3-regular graphs is enough to prove it for all graphs.