Groups, graphs and mutually unbiased bases

A new characteristic free approach to constructing large sets of mutually unbiased bases in Hilbert space is developed. We associate with a seed set of bases a finite subgroup of which defines a strongly regular graph. Large sets of mutually unbiased bases are obtained as the cliques of the graph.

Clock synchronization state graphs based on clock precision difference

Consistent and stable global states of clock synchronization are very important in distributed and parallel systems. This paper presents an innovative strategy and method to obtain stable global clock synchronization state graphs in asynchronous Internet environments. Our model will introduce the concept of clock precision difference as a means to evaluate running states of all clocks in this system and make this system  self-adaptive well. Finally, we introduce the concept of clock precision difference into global states analysis of clock synchronization and construct clock synchronization state graphs in order to evaluate distributed clock synchronization states. We also present detailed simulations of the strategy and mathematical analysis used on real Internet environments.

Intersection and partition graphs

This thesis reviews previous work done on both general partition graphs and existential partition graphs, which are a generalised form of general partition graphs, and extends some of the results.

Malware variant detection using similarity search over sets of control flow graphs

Static detection of polymorphic malware variants plays an important role to improve system security. Control flow has shown to be an effective characteristic that represents polymorphic malware instances. In our research, we propose a similarity search of malware using novel distance metrics of malware signatures. We describe a malware signature by the set of control flow graphs the malware contains. We propose two approaches and use the first to perform pre-filtering. Firstly, we use a distance metric based on the distance between feature vectors. The feature vector is a decomposition of the set of graphs into either fixed size k-sub graphs, or q-gram strings of the high-level source after decompilation. We also propose a more effective but less computationally efficient distance metric based on the minimum matching distance. The minimum matching distance uses the string edit distances between programs' decompiled flow graphs, and the linear sum assignment problem to construct a minimum sum weight matching between two sets of graphs. We implement the distance metrics in a complete malware variant detection system. The evaluation shows that our approach is highly effective in terms of a limited false positive rate and our system detects more malware variants when compared to the detection rates of other algorithms.

The L(2,1)-labelling problem for cubic Cayley graphs on dihedral groups

A k-L(2,1)-labelling of a graph G is a mapping f:V(G)→{0,1,2,…,k} such that |f(u)−f(v)|≥2 if uv∈E(G) and f(u)≠f(v) if u,v are distance two apart. The smallest positive integer k such that G admits a k-L(2,1)-labelling is called the λ-number of G. In this paper we study this quantity for cubic Cayley graphs (other than the prism graphs) on dihedral groups, which are called brick product graphs or honeycomb toroidal graphs. We prove that the λ-number of such a graph is between 5 and 7, and moreover we give a characterisation of such graphs with λ-number 5.

Power graphs : a survey

This article gives a survey of all results on the power graphs of groups and semigroups obtained in the literature. Various conjectures due to other authors, questions and open problems are also included.

Incidence semirings of graphs and visible bases

We consider the incidence semirings of graphs and prove that every incidence semiring has convenient visible bases for its right ideals and for its left ideals, and that these visible bases can be used to determine the weights of all right ideals that have maximum weight and all left ideals that have maximum weight. ©2013 Australian Mathematical Publishing Association Inc..

Stabilizing high-dimensional prediction models using feature graphs

We investigate feature stability in the context of clinical prognosis derived from high-dimensional electronic medical records. To reduce variance in the selected features that are predictive, we introduce Laplacian-based regularization into a regression model. The Laplacian is derived on a feature graph that captures both the temporal and hierarchic relations between hospital events, diseases, and interventions. Using a cohort of patients with heart failure, we demonstrate better feature stability and goodness-of-fit through feature graph stabilization.

On the kidney exchange problem: cardinality constrained cycle and chain problems on directed graphs: a survey of integer programming approaches

The Kidney Exchange Problem (KEP) is a combinatorial optimization problem and has attracted the attention from the community of integer programming/combinatorial optimisation in the past few years. Defined on a directed graph, the KEP has two variations: one concerns cycles only, and the other, cycles as well as chains on the same graph. We call the former a Cardinality Constrained Multi-cycle Problem (CCMcP) and the latter a Cardinality Constrained Cycles and Chains Problem (CCCCP). The cardinality for cycles is restricted in both CCMcP and CCCCP. As for chains, some studies in the literature considered cardinality restrictions, whereas others did not. The CCMcP can be viewed as an Asymmetric Travelling Salesman Problem that does allow subtours, however these subtours are constrained by cardinality, and that it is not necessary to visit all vertices. In existing literature of the KEP, the cardinality constraint for cycles is usually considered to be small (to the best of our knowledge, no more than six). In a CCCCP, each vertex on the directed graph can be included in at most one cycle or chain, but not both. The CCMcP and the CCCCP are interesting and challenging combinatorial optimization problems in their own rights, particularly due to their similarities to some travelling salesman- and vehicle routing-family of problems. In this paper, our main focus is to review the existing mathematical programming models and solution methods in the literature, analyse the performance of these models, and identify future research directions. Further, we propose a polynomial-sized and an exponential-sized mixed-integer linear programming model, discuss a number of stronger constraints for cardinality-infeasible-cycle elimination for the latter, and present some preliminary numerical results.

Ideal bases in constructions defined by directed graphs

The present article continues the investigation of visible ideal bases in constructions defined using directed graphs. Our main theorem establishes that, for every balanced digraph D and each idempotent semiring R with 1, the incidence semiring ID(R) of the digraph D has a convenient visible ideal basis BD(R). It also shows that the elements of BD(R) can always be used to generate two-sided ideals with the largest possible weight among the weights of all two-sided ideals in the incidence semiring.

An embedding scheme for detecting anomalous block structured graphs

Graph-based anomaly detection plays a vital role in various application domains such as network intrusion detection, social network analysis and road traffic monitoring. Although these evolving networks impose a curse of dimensionality on the learning models, they usually contain structural properties that anomaly detection schemes can exploit. The major challenge is finding a feature extraction technique that preserves graph structure while balancing the accuracy of the model against its scalability. We propose the use of a scalable technique known as random projection as a method for structure aware embedding, which extracts relational properties of the network, and present an analytical proof of this claim. We also analyze the effect of embedding on the accuracy of one-class support vector machines for anomaly detection on real and synthetic datasets. We demonstrate that the embedding can be effective in terms of scalability without detrimental influence on the accuracy of the learned model.