Determinants of grids, tori, cylinders and M\"{o}bius ladders

Recently, Bie\~{n} [A. Bie\~{n}, The problem of singularity for planar grids, Discrete Math. 311 (2011), 921--931] obtained a recursive formula for the determinant of a grid. Also, recently, Pragel [D. Pragel, Determinants of box products of paths, Discrete Math. 312 (2012), 1844--1847], independently, obtained an explicit formula for this determinant. In this paper, we give a short proof for this problem. Furthermore, applying the same technique, we get explicit formulas for the determinant of a torus, a cylinder, and a M\"{o}bius ladder

On fully split lacunary polynomials in finite fields

We estimate the number of possible types degree patterns of $k$-lacunary polynomials of degree $t < p$ which split completely modulo $p$. The result is based on a combination of a bound on the number of zeros of lacunary polynomials with some graph theory arguments.Comment: 8 pages. Bull. Polish Acad. Sci. Math., to appea

Counting surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group with motivations from string theory and QFT

Graphs embedded into surfaces have many important applications, in particular, in combinatorics, geometry, and physics. For example, ribbon graphs and their counting is of great interest in string theory and quantum field theory (QFT). Recently, Koch, Ramgoolam, and Wen [Nuclear Phys.\,B {\bf 870} (2013), 530--581] gave a refined formula for counting ribbon graphs and discussed its applications to several physics problems. An important factor in this formula is the number of surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group. The aim of this paper is to give an explicit and practical formula for the number of such epimorphisms. As a consequence, we obtain an `equivalent' form of the famous Harvey's theorem on the cyclic groups of automorphisms of compact Riemann surfaces. Our main tool is an explicit formula for the number of solutions of restricted linear congruence recently proved by Bibak et al. using properties of Ramanujan sums and of the finite Fourier transform of arithmetic functions

Contributions at the Interface Between Algebra and Graph Theory

In this thesis, we make some contributions at the interface between algebra and graph theory. In Chapter 1, we give an overview of the topics and also the definitions and preliminaries. In Chapter 2, we estimate the number of possible types degree patterns of k-lacunary polynomials of degree t < p which split completely modulo p. The result is based on a rather unusual combination of two techniques: a bound on the number of zeros of lacunary polynomials and a bound on the so-called domination number of a graph. In Chapter 3, we deal with the determinant of bipartite graphs. The nullity of a graph G is the multiplicity of 0 in the spectrum of G. Nullity of a (molecular) graph (e.g., a bipartite graph corresponding to an alternant hydrocarbon) has important applications in quantum chemistry and Huckel molecular orbital (HMO) theory. A famous problem, posed by Collatz and Sinogowitz in 1957, asks to characterize all graphs with positive nullity. Clearly, examining the determinant of a graph is a way to attack this problem. In this Chapter, we show that the determinant of a bipartite graph with at least two perfect matchings and with all cycle lengths divisible by four, is zero. In Chapter 4, we first introduce an application of spectral graph theory in proving trigonometric identities. This is a very simple double counting argument that gives very short proofs for some of these identities (and perhaps the only existed proof in some cases!). In the rest of Chapter 4, using some properties of the well-known Chebyshev polynomials, we prove some theorems that allow us to evaluate the number of spanning trees in join of graphs, Cartesian product of graphs, and nearly regular graphs. In the last section of Chapter 4, we obtain the number of spanning trees in an (r,s)-semiregular graph and its line graph. Note that the same results, as in the last section, were proved by I. Sato using zeta functions. But our proofs are much shorter based on some well-known facts from spectral graph theory. Besides, we do not use zeta functions in our arguments. In Chapter 5, we present the conclusion and also some possible projects

On an almost-universal hash function family with applications to authentication and secrecy codes

Universal hashing, discovered by Carter and Wegman in 1979, has many important applications in computer science. MMH$^*$, which was shown to be $\Delta$-universal by Halevi and Krawczyk in 1997, is a well-known universal hash function family. We introduce a variant of MMH$^*$, that we call GRDH, where we use an arbitrary integer $n>1$ instead of prime $p$ and let the keys $\mathbf{x}=\langle x_1, \ldots, x_k \rangle \in \mathbb{Z}_n^k$ satisfy the conditions $\gcd(x_i,n)=t_i$ ($1\leq i\leq k$), where $t_1,\ldots,t_k$ are given positive divisors of $n$. Then via connecting the universal hashing problem to the number of solutions of restricted linear congruences, we prove that the family GRDH is an $\varepsilon$-almost-$\Delta$-universal family of hash functions for some $\varepsilon<1$ if and only if $n$ is odd and $\gcd(x_i,n)=t_i=1$ $(1\leq i\leq k)$. Furthermore, if these conditions are satisfied then GRDH is $\frac{1}{p-1}$-almost-$\Delta$-universal, where $p$ is the smallest prime divisor of $n$. Finally, as an application of our results, we propose an authentication code with secrecy scheme which strongly generalizes the scheme studied by Alomair et al. [{\it J. Math. Cryptol.} {\bf 4} (2010), 121--148], and [{\it J.UCS} {\bf 15} (2009), 2937--2956].Comment: International Journal of Foundations of Computer Science, to appea

Restricted linear congruences

In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence $a_1x_1+\cdots +a_kx_k\equiv b \pmod{n}$, with $\gcd(x_i,n)=t_i$ ($1\leq i\leq k$), where $a_1,t_1,\ldots,a_k,t_k, b,n$ ($n\geq 1$) are arbitrary integers. As a consequence, we derive necessary and sufficient conditions under which the above restricted linear congruence has no solutions. The number of solutions of this kind of congruence was first considered by Rademacher in 1925 and Brauer in 1926, in the special case of $a_i=t_i=1$ $(1\leq i \leq k)$. Since then, this problem has been studied, in several other special cases, in many papers; in particular, Jacobson and Williams [{\it Duke Math. J.} {\bf 39} (1972), 521--527] gave a nice explicit formula for the number of such solutions when $(a_1,\ldots,a_k)=t_i=1$ $(1\leq i \leq k)$. The problem is very well-motivated and has found intriguing applications in several areas of mathematics, computer science, and physics, and there is promise for more applications/implications in these or other directions.Comment: Journal of Number Theory, to appea