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 Investigating Hamilton cycles in cubic graphs using logistic regression Alex Newcombe, supervised by Prof. Jerzy Filar.