Extrema of graph eigenvalues

In 1993 Hong asked what are the best bounds on the $k$'th largest eigenvalue $\lambda_{k}(G)$ of a graph $G$ of order $n$. This challenging question has never been tackled for any $2<k<n$. In the present paper tight bounds are obtained for all $k>2,$ and even tighter bounds are obtained for the $k$'th largest singular value $\lambda_{k}^{\ast}(G).$ Some of these bounds are based on Taylor's strongly regular graphs, and other on a method of Kharaghani for constructing Hadamard matrices. The same kind of constructions are applied to other open problems, like Nordhaus-Gaddum problems of the kind: How large can $\lambda_{k}(G)+\lambda_{k}(\bar{G})$ be$?$ These constructions are successful also in another open question: How large can the Ky Fan norm $\lambda_{1}^{\ast}(G)+...+\lambda_{k}^{\ast }(G)$ be $?$ Ky Fan norms of graphs generalize the concept of graph energy, so this question generalizes the problem for maximum energy graphs. In the final section, several results and problems are restated for $(-1,1)$-matrices, which seem to provide a more natural ground for such research than graphs. Many of the results in the paper are paired with open questions and problems for further study.Comment: 32 page

Latin Squares and Their Applications to Cryptography

A latin square of order-n is an n x n array over a set of n symbols such that every symbol appears exactly once in each row and exactly once in each column. Latin squares encode features of algebraic structures. When an algebraic structure passes certain latin square tests , it is a candidate for use in the construction of cryptographic systems. A transversal of a latin square is a list of n distinct symbols, one from each row and each column. The question regarding the existence of transversals in latin squares that encode the Cayley tables of finite groups is far from being resolved and is an area of active investigation. It is known that counting the pairs of permutations over a Galois field ��pd whose point-wise sum is also a permutation is equivalent to counting the transversals of a latin square that encodes the addition group of ��pd. We survey some recent results and conjectures pertaining to latin squares and transversals. We create software tools that generate latin squares and count their transversals. We confirm previous results that cyclic latin squares of prime order-p possess the maximum transversal counts for 3 ≤ p ≤ 9. Furthermore, we create a new algorithm that uses these prime order-p cyclic latin squares as building blocks to construct super-symmetric latin squares of prime power order-pd with d \u3e 0; using this algorithm we accurately predict that super-symmetric latin squares of order-pd possess the confirmed maximum transversal counts for 3 ≤ pd ≤ 9 and the estimated lower bound on the maximum transversal counts for 9 \u3c pd ≤ 17. Also, we give some conjectures regarding the number of transversals in a super-symmetric latin square. Lastly, we use the super-symmetric latin square for the additive group of the Galois field (��32, +) to create a simplified version of Grøstl, an iterated hash function, where the compression function is built from two fixed, large, distinct permutations

Ryser Type Conditions for Extending Colorings of Triples

In 1951, Ryser showed that an $n\times n$ array $L$ whose top left $r\times s$ subarray is filled with $n$ different symbols, each occurring at most once in each row and at most once in each column, can be completed to a latin square of order $n$ if and only if the number of occurrences of each symbol in $L$ is at least $r+s-n$. We prove a Ryser type result on extending partial coloring of 3-uniform hypergraphs. Let $X,Y$ be finite sets with $X\subsetneq Y$ and $|Y|\equiv 0 \pmod 3$. When can we extend a (proper) coloring of $\lambda \binom{X}{3}$ (all triples on a ground set $X$, each one being repeated $\lambda$ times) to a coloring of $\lambda \binom{Y}{3}$ using the fewest possible number of colors? It is necessary that the number of triples of each color in $\binom{X}{3}$ is at least $|X|-2|Y|/3$. Using hypergraph detachments (Combin. Probab. Comput. 21 (2012), 483--495), we establish a necessary and sufficient condition in terms of list coloring complete multigraphs. Using H\"aggkvist-Janssen's bound (Combin. Probab. Comput. 6 (1997), 295--313), we show that the number of triples of each color being at least $|X|/2-|Y|/6$ is sufficient. Finally we prove an Evans type result by showing that if $|Y|\geq 3|X|$, then any $q$-coloring of any subset of $\lambda \binom{X}{3}$ can be embedded into a $\lambda\binom{|Y|-1}{2}$-coloring of $\lambda \binom{Y}{3}$ as long as $q\leq \lambda \binom{|Y|-1}{2}-\lambda \binom{|X|}{3}/\lfloor{|X|/3}\rfloor$.Comment: 10 page

A historical perspective of the theory of isotopisms

In the middle of the twentieth century, Albert and Bruck introduced the theory of isotopisms of non-associative algebras and quasigroups as a generalization of the classical theory of isomorphisms in order to study and classify such structures according to more general symmetries. Since then, a wide range of applications have arisen in the literature concerning the classification and enumeration of different algebraic and combinatorial structures according to their isotopism classes. In spite of that, there does not exist any contribution dealing with the origin and development of such a theory. This paper is a first approach in this regard.Junta de Andalucí

Linear spaces with many small lines

AbstractIn this paper some of the work in linear spaces in which most of the lines have few points is surveyed. This includes existence results, blocking sets and embeddings. Also, it is shown that any linear space of order v can be embedded in a linear space of order about 13v in which there are no lines of size 2

Completing Incomplete Commutative Latin Squares With Prescribed Diagonals

The main diagonal and the upper left-hand r × r square of an n × n array contain symbols, the remaining cells are empty. We give simple necessary and sufficient conditions for completing the array to a commutative Latin square. We apply these results to give a short proof of Cruse's theorem, and an embedding theorem for half-idempotent commutative Latin squares