# Center for Teaching and Learning

## SCCUR 2004 Sessions at Whittier College

 Cover Pebbling Multipartite Graphs Marlene M. Merchain (Sponsored by the Mathematical Association of America“s REU in Mathematics at California Lutheran University; funded by NSA and NSF; Mentor: Dr. Cynthia J. Wyels) Consider a distribution of pebbles to the vertices of a graph. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. We ask whether, using these moves, it is possible to “cover“ the graph, i.e. to place a pebble on every vertex of the graph (simultaneously). The cover pebbling number of a graph is the minimum number of pebbles needed to cover the graph, regardless of how the pebbles were initially distributed. We explore the cover pebbling numbers of multipartite graphs, determining lower bounds for cover pebbling numbers in many instances of multipartite graphs. The Decline of Irish Monasticism Josh Carr, History (Dr. Paul Hanson) The independent development of Christianity in Ireland between the fifth and sixth centuries produced a distinct and significant form of monasticism that eventually extended into northern England. However, Irish Monasticism could not withstand the competition from the new form of ecclesiastical organization and authority introduced by Papal missionaries from Rome in southern England. Though the Irish Monks helped preserve many ancient biblical texts, they were forced to capitulate to the dominance of Papal supremacy in England in the 7 th century. Sponsored by NSF and the Summer REU in Mathematics at California Lutheran University; funded by NSA and NSF Cover Pebbling Number for Cycles and Graham’s Conjecture Victor M. Moreno, California State University, Channel Islands (Dr. Cynthia Wyels) Consider a distribution of pebbles to the vertices of a graph. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. We ask whether, using these moves, it is possible to “cover“ the graph, i.e. to place a pebble on every vertex of the graph (simultaneously). The cover pebbling number of a graph is the minimum number of pebbles needed to cover the graph, regardless of how the pebbles were initially distributed. In this paper we explore the cover pebbling number of cycle graphs. Graham“s conjecture is perhaps the most famous open conjecture in pebbling; it concerns pebbling numbers of products of graphs. We investigate Graham’s conjecture as modified for cover pebbling. Sponsored by NSF and the Summer REU in Mathematics at Cal State San Bernardino 2004 The Supercrossing Index of Torus Knots Ryan Petitfils, Mathematics Major, CLU (Advisor: Dr. Rolland Trapp, CSUSB) Knot Theory, which is part of an area of mathematics known as Topology, is studied in order to describe properties of knots as well as invariants that show if two knots are equivalent. Knots can be configured with many types of objects, such as a rope or sticks. The crossing index of a given knot is the least possible number of crossings over any configuration of that knot. The Supercrossing Index of a knot, on the other hand, represents the minimum of the greatest number of crossings possible taken over any configuration of a knot. A Torus Knot is any knot that can be put onto a torus, a donut shaped surface, such that no two strands cross on the surface of that torus. The minimal stick representations of Torus Knots are used to find bounds for the Supercrossing Index. Through my research, I greatly improved the Supercrossing Index upper bound for a certain type of Torus Knot.