1. The mathematics of graphs A graph has many representations, the simplest being a collection of dots (vertices) and lines (edges). Below is a cubic graph (3 edges connected to each vertex) with its adjacency matrix representation. An important class of cubic graphs are non-bridge graphs, these graphs have no pairs of vertices that are connected by just one, non-repeating path (the figure below is a bridge graph, the one above is a non-bridge graph). Non-bridge, cubic graphs are the graphs considered for this investigation. The Hamilton cycle problem (HCP) Given a graph, determine whether or not a path exists that visits every vertex only once and returns to the starting vertex. It is a deceptively simple question, do the graphs in the section above have a Hamilton cycle? HCP for cubic graphs is directly connected to a very fundamental, unsolved problem called P vs NP. It is the problem of determining if a certain difficult class of problems can possess efficient algorithms to solve them (for an in-depth description of this famous problem see [2]). The figure below has two graphs, each with one of their Hamilton cycles highlighted in red. Functions of graphs The adjacency matrix representation of a graph opens it up to tools from linear algebra. We define a function of a graph to be some function of its adjacency matrix or some form of descriptor of that graph. Two examples considered in this investigation are: Second largest eigenvalue of the adjacency matrix – this value describes the average connectivity present in the graph (think of a barbell-like structure with a thin centre connected to two large components versus a circular structure where all vertices are ‘near’). Resistance matrix – Has entries which describe the effective resistance between two vertices. If two vertices have many possible non-repeating paths between them, then the effective resistance between them is low, similarly few possible paths means a high effective resistance. Regions of non-bridge non-Hamiltonian cubic graphs An observation resulting from this investigation is a clustering effect that occurs with some functions when applied to cubic graphs (see the figure above). We may plot all non-bridge, cubic graphs on say N vertices with respect to two functions. Note that this is not an easy task as even for a moderate N=24 vertices, there are already 119,709,267 cubic graphs. Regardless, we observe that for particular functions, non- bridge non-Hamiltonian graphs appear as clusters, mixed in amongst the Hamiltonian graphs. We compare plots to others using the same functions, but now considering all cubic graphs on N+2 vertices, then N+4 vertices and so on. The clusters of non-bridge non-Hamiltonian graphs can be seen to carry-over and become larger and more dense as they do. Investigating the regions with logistic regression Using a statistical model constructed from the functions where the clustering effect was observed, we can approximate the region where most of the non-bridge non-Hamiltonian graphs reside. In particular because of the carry-over effect, the model can be used to approximate this region for cubic graphs on more vertices. From the model we construct a test that determines if a cubic graph lies inside or outside of this region. Then using a random cubic graph generating algorithm, we determine the approximate proportion of non-bridge non-Hamiltonian graphs inside the region, versus outside the region. The main result of the investigation is the ability to identify these regions where the concentration of non-bridge non-Hamiltonian cubic graphs is much greater. The figure below first shows one of the approximated regions, then the next two plots are of a random sample of cubic graphs on a larger number of vertices. The majority of the non-bridge non-Hamiltonian graphs still reside inside the region. Future research An avenue for future research is an analysis of the density of Hamiltonian graphs outside the regions. There may be a discernible difference between the region where most Hamiltonian graphs reside and the region where most non-bridge non-Hamiltonian cubic graphs reside that can be characterized. [1] – Borkar S., Ejov V., Filar J., Nyugen G., ‘Hamilton Cycle Problem and Markov Chains’, 2012, Springer. [2] – Goldreich O., ‘P, NP, and NP-Completeness: The Basics of Complexity Theory’, 2010, Cambridge University Press. Aim of the investigation We aim to investigate trends in the values of functions of non-bridge cubic graphs, with respect to whether or not the graph is non-bridge, non-Hamiltonian. Then to be able to probabilistically test any cubic graph and decide how likely it is to be a non-bridge non-Hamiltonian graph. The idea originated from an observation by Filar (described in detail in [1]), which shows a very regular structure that is formed due to the number of cycles of different lengths that are present in the cubic graphs. 9 8 6 4 10 5 3 2 7 1 Investigating Hamilton cycles in cubic graphs using logistic regression Alex Newcombe, supervised by Prof. Jerzy Filar.