Structure and properties of maximal outerplanar graphs.

Outerplanar graphs are planar graphs that have a plane embedding in which each vertex lies on the boundary of the exterior region. An outerplanar graph is maximal outerplanar if the graph obtained by adding an edge is not outerplanar. Maximal outerplanar graphs are also known as triangulations of polygons. The spine of a maximal outerplanar graph G is the dual graph of G without the vertex that corresponds to the exterior region. In this thesis we study metric properties involving geodesic intervals, geodetic sets, Steiner sets, different concepts of boundary, and also relationships between the independence numbers and domination numbers of maximal outerplanar graphs and their spines. In Chapter 2 we find an extension of a result by Beyer, et al. [3] that deals with Hamiltonian degree sequences in maximal outerplanar graphs. In Chapters 3 and 4 we give sharp bounds relating the independence number and domination number, respectively, of a maximal outerplanar graph to those of its spine. In Chapter 5 we discuss the boundary, contour, eccentricity, periphery, and extreme set of a graph. We give a characterization of the boundary of maximal outerplanar graphs that involves the degrees of vertices. We find properties that characterize the contour of a maximal outerplanar graph. The other main result of this chapter gives characterizations of graphs induced by the contour and by the periphery of a maximal outerplanar graph. In Chapter 6 we show that the generalized intervals in a maximal outerplanar graph are convex. We use this result to characterize geodetic sets in maximal outerplanar graphs. We show that every Steiner set in a maximal outerplanar graph is a geodetic set and also show some differences between these types of sets. We present sharp bounds for geodetic numbers and Steiner numbers of maximal outerplanar graphs

Quirks of Stirling\u27s Approximation

Stirling\u27s approximation to ln n! is typically introduced to physical chemistry students as a step in the derivation-of the statistical expression for the entropy. However, naive application of this approximation leads to incorrect conclusions. In this article, the problem is first illustrated using a familiar toy model example, the two-state system of N classical spins, where it is shown that two different physical situations lead to the same computed value of the entropy. Retention of additional terms in the approximation of the factorial is required to yield an accurate expression for the statistical weight of the most probable configuration in such model systems, but generates only a little extra accuracy in entropy calculations, and then only in the limit of very small numbers of particles. Additionally, inclusion of these terms makes the entropy nonextensive. We show here that, in the standard derivation of the entropy of the microcanonical ensemble, it is the freedom to allow the ensemble size to be infinite that makes the Boltzmann entropy expression S = k(B) lnW exact, a fact that is not widely understood