Rings, Group Rings, and Their Graphs

We associate some graphs to a ring R and we investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of the graphs associated to R. Let Z(R) be the set of zero-divisors of R. We define an undirected graph ᴦ(R) with nonzero zero-divisors as vertices and distinct vertices x and y are adjacent if xy=0 or yx=0. We investigate the Isomorphism Problem for zero-divisor graphs of group rings RG. Let Sk denote the sphere with k handles, where k is a non-negative integer, that is, Sk is an oriented surface of genus k. The genus of a graph is the minimal integer n such that the graph can be embedded in Sn. The annihilating-ideal graph of R is defined as the graph AG(R) with the set of ideals with nonzero annihilators as vertex such that two distinct vertices I and J are adjacent if IJ=(0). We characterize Artinian rings whose annihilating-ideal graphs have finite genus. Finally, we extend the definition of the annihilating-ideal graph to non-commutative rings.

Educational Leadership: Exploring the Influence of Managed and Living Systems in Educational Leadership Discourse

A conceptual analysis of educational leadership explored the influence of managed and living systems on 21st century leadership discourse. Drawing on a detailed understanding of managed and living systems theory compiled from the work of Capra (2002), Morgan (1997), Mitchell and Sackney (2009), and Wheatley (2007), this study draws attention to the managed systems systemic concepts of efficiency, control, and standardization, and the living systems concepts of collaboration, shared meaning, change, and interconnection as markers of systems theory that find resonance within leadership literature. Using these systemic concepts as a framework, this study provides important insights into the espousal of managed and living systems concepts within the leadership discourse.

Backbone Colouring of Graphs

Consider an undirected graph G and a subgraph of G, H. A q-backbone k-colouring of (G,H) is a mapping f: V(G) {1, 2, ..., k} such that G is properly coloured and for each edge of H, the colours of its endpoints differ by at least q. The minimum number k for which there is a backbone k-colouring of (G,H) is the backbone chromatic number, BBCq(G,H). It has been proved that backbone k-colouring of (G,T) is at most 4 if G is a connected C4-free planar graph or non-bipartite C5-free planar graph or Cj-free, j∈{6,7,8} planar graph without adjacent triangles. In this thesis we improve the results mentioned above and prove that 2-backbone k-colouring of any connected planar graphs without adjacent triangles is at most 4 by using a discharging method. In the second part of this thesis we further improve these results by proving that for any graph G with χ(G) ≥ 4, BBC(G,T) = χ(G). In fact, we prove the stronger result that a backbone tree T in G exists, such that ∀ uv ∈ T, |f(u)-f(v)|=2 or |f(u)-f(v)| ≥ k-2, k = χ(G). For the case that G is a planar graph, according to Four Colour Theorem, χ(G) = 4; so, BBC(G,T) = 4.

Working Together: Librarian and Student Collaboration Through Active Learning in a Library Eclassroom

Active learning strategies based on several learning theories were incorporated during instruction sessions for second year Biological Sciences students. The instructional strategies described in this paper are based primarily on sociocultural and collaborative learning theory, with the goal being to expand the relatively small body of literature currently available that discusses the application of these learning theories to library instruction. The learning strategies employed successfully involved students in the learning process ensuring that the experiences were appropriate and effective. The researchers found that, as a result of these strategies (e.g. teaching moments based on the emerging needs of students) students’ interest in learning information literacy was increased and students interacted with information given to them as well as with their peers. Collaboration between the Librarians, Co-op Student and Senior Lab Instructor helped to enhance the learning experience for students and also revealed new aspects of the active learning experiences. The primary learning objective, which was to increase the students’ information skills in the Biological Sciences, was realized. The advantages of active learning were realized by both instructors and students. Advantages for students attained during these sessions include having their diverse learning styles addressed; increased interaction with and retention of information; increased responsibility for their own learning; the opportunity to value not only the instructors, but also themselves and their peers as sources of authority and knowledge; improved problem solving abilities; increased interest and opportunities for critical thinking, as a result of the actively exchanging information in a group. The primary advantage enjoyed by the instructors was the opportunity to collaborate with colleagues to reduce the preparation required to create effective library instruction sessions. Opportunities for further research were also discovered, including the degree to which “social loafing” plays a role in collaborative, active learning.