Isoperimetric numbers of Cayley graphs arising from generalized dihedral group

Let n, x be positive integers satisfying 1 < x < n. Let Hn,x be a group admitting a presentation of the form 〈a, b | a n = b 2 = (ba) x = 1〉. When x = 2 the group Hn,x is the familiar dihedral group, D2n. Groups of the form Hn,x will be referred to as generalized dihedral groups. It is possible to associate a cubic Cayley graph to each such group, and we consider the problem of finding the isoperimetric number, i(G), of these graphs. In section two we prove some propositions about isoperimetric numbers of regular graphs. In section three the special cases when x = 2, 3 are analyzed. The former case is solved completely. An upper bound, based on an analysis of the cycle structure of the graph, is given in the latter case. Generalizations of these results are provided in section four. The indices of these graphs are calculated in section five, and a lower bound on i(G) is obtained as a result. We conclude with several conjectures suggested by the results from earlier sections

Cheeger Constants of Platonic Graphs

Abstract. The Platonic graphs πn arise in several contexts, most simply as a quotient of certain Cayley graphs associated to the projective special linear groups. We show that when n = p is prime, πn can be viewed as a complete multigraph in which each vertex is itself a wheel on n + 1 vertices. We prove a similar structure theorem for the case of an arbitrary prime power. These theorems are then used to obtain new upper bounds on the Cheeger constants of these graphs. These results lead immediatley to similar results for Cayley graphs of the group P SL(2, Zn). 1

A Decomposition Theorem for Cayley Graphs of Picard Group Quotients

Abstract. The Picard group is defined as Γ = SL(2, Z[i]); the ring of 2 × 2 matrices with Gaussian integer entries and determinant one. We consider certain graphs associated to quotients Γ/Γ(p) where p is a prime congruent to three mod four and Γ(p) is the congruence subgroup of level p. We prove a decomposition theorem on the vertices of these graphs, and use this decomposition to derive upper and lower bounds on their isoperimetric numbers. 1

The mathematics of various entertaining subjects

Volume 1 : The history of mathematics is filled with major breakthroughs resulting from solutions to recreational problems. Problems of interest to gamblers led to the modern theory of probability, for example, and surreal numbers were inspired by the game of Go. Yet even with such groundbreaking findings and a wealth of popular-level books exploring puzzles and brainteasers, research in recreational mathematics has often been neglected. The Mathematics of Various Entertaining Subjects brings together authors from a variety of specialties to present fascinating problems and solutions in recreational mathematics. Contributors to the book show how sophisticated mathematics can help construct mazes that look like famous people, how the analysis of crossword puzzles has much in common with understanding epidemics, and how the theory of electrical circuits is useful in understanding the classic Towers of Hanoi puzzle. The card game SET is related to the theory of error-correcting codes, and simple tic-tac-toe takes on a new life when played on an affine plane. Inspirations for the book's wealth of problems include board games, card tricks, fake coins, flexagons, pencil puzzles, poker, and so much more. Looking at a plethora of eclectic games and puzzles, The Mathematics of Various Entertaining Subjects is sure to entertain, challenge, and inspire academic mathematicians and avid math enthusiasts alike. Volume 2 : This latest volume gathers together the top experts in recreational math and presents a compelling look at board games, card games, dice, toys, computer games, and much more. The book is divided into five parts: puzzles and brainteasers, geometry and topology, graph theory, games of chance, and computational complexity. Readers will discover what origami, roulette wheels, and even the game of Trouble can teach about math. Essays contain new results, and the contributors include short expositions on their topic's background, providing a framework for understanding the relationship between serious mathematics and recreational games. Mathematical areas explored include combinatorics, logic, graph theory, linear algebra, geometry, topology, computer science, operations research, probability, game theory, and music theory. Investigating an eclectic mix of games and puzzles, The Mathematics of Various Entertaining Subjects is sure to entertain, challenge, and inspire academic mathematicians and avid math enthusiasts alike

Lower Bounds on the Cheeger Constants of Highly Connected Regular Graphs

Abstract. We develop a method for obtaining lower bounds on the Cheeger constants of certain highly connected graphs. We then apply this technique to obtain new lower bounds on the Cheeger constants of two important families of graphs. Finally, we discuss the relevance of our bounds to determining the integrity of these graphs. 1