Optimal monomial quadratization for ODE systems

Quadratization problem is, given a system of ODEs with polynomial right-hand side, transform the system to a system with quadratic right-hand side by introducing new variables. Such transformations have been used, for example, as a preprocessing step by model order reduction methods and for transforming chemical reaction networks. We present an algorithm that, given a system of polynomial ODEs, finds a transformation into a quadratic ODE system by introducing new variables which are monomials in the original variables. The algorithm is guaranteed to produce an optimal transformation of this form (that is, the number of new variables is as small as possible), and it is the first algorithm with such a guarantee we are aware of. Its performance compares favorably with the existing software, and it is capable to tackle problems that were out of reach before

Adjacency matrices of polarity graphs and of other C4-free graphs of large size

In this paper we give a method for obtaining the adjacency matrix of a simple polarity graph $G_q$ from a projective plane PG(2,q), where q is a prime power. Denote by ex(n;C₄) the maximum number of edges of a graph on n vertices and free of squares C₄. We use the constructed graphs $G_q$ to obtain lower bounds on the extremal function ex(n;C₄), for some n < q² + q + 1. In particular, we construct a C₄-free graph on n = q² − √q vertices and 1/2q(q²−1) − 1/2√q(q − 1) edges, for a square prime power q

Adjacency matrices of polarity graphs and of other C4-free graphs of large size

In this paper we give a method for obtaining the adjacency matrix of a simple polarity graph $G_q$ from a projective plane PG(2,q), where q is a prime power. Denote by ex(n;C₄) the maximum number of edges of a graph on n vertices and free of squares C₄. We use the constructed graphs $G_q$ to obtain lower bounds on the extremal function ex(n;C₄), for some n < q² + q + 1. In particular, we construct a C₄-free graph on n = q² − √q vertices and 1/2q(q²−1) − 1/2√q(q − 1) edges, for a square prime power q

On the structure of graphs without short cycles

The objective of this thesis is to study cages, constructions and properties of such families of graphs. For this, the study of graphs without short cycles plays a fundamental role in order to develop some knowledge on their structure, so we can later deal with the problems on cages. Cages were introduced by Tutte in 1947. In 1963, Erdös and Sachs proved that (k, g) -cages exist for any given values of k and g. Since then, large amount of research in cages has been devoted to their construction. In this work we study structural properties such as the connectivity, diameter, and degree regularity of graphs without short cycles. In some sense, connectivity is a measure of the reliability of a network. Two graphs with the same edge-connectivity, may be considered to have different reliabilities, as a more refined index than the edge-connectivity, edge-superconnectivity is proposed together with some other parameters called restricted connectivities. By relaxing the conditions that are imposed for the graphs to be cages, we can achieve more refined connectivity properties on these families and also we have an approach to structural properties of the family of graphs with more restrictions (i.e., the cages). Our aim, by studying such structural properties of cages is to get a deeper insight into their structure so we can attack the problem of their construction. By way of example, we studied a condition on the diameter in relation to the girth pair of a graph, and as a corollary we obtained a result guaranteeing restricted connectivity of a special family of graphs arising from geometry, such as polarity graphs. Also, we obtained a result proving the edge superconnectivity of semiregular cages. Based on these studies it was possible to develop the study of cages. Therefore obtaining a relevant result with respect to the connectivity of cages, that is, cages are k/2-connected. And also arising from the previous work on girth pairs we obtained constructions for girth pair cages that proves a bound conjectured by Harary and Kovács, relating the order of girth pair cages with the one for cages. Concerning the degree and the diameter, there is the concept of a Moore graph, it was introduced by Hoffman and Singleton after Edward F. Moore, who posed the question of describing and classifying these graphs. As well as having the maximum possible number of vertices for a given combination of degree and diameter, Moore graphs have the minimum possible number of vertices for a regular graph with given degree and girth. That is, any Moore graph is a cage. The formula for the number of vertices in a Moore graph can be generalized to allow a definition of Moore graphs with even girth (bipartite Moore graphs) as well as odd girth, and again these graphs are cages. Thus, Moore graphs give a lower bound for the order of cages, but they are known to exist only for very specific values of k, therefore it is interesting to study how far a cage is from this bound, this value is called the excess of a cage. We studied the excess of graphs and give a contribution, in the sense of the work of Biggs and Ito, relating the bipartition of girth 6 cages with their orders. Entire families of cages can be obtained from finite geometries, for example, the graphs of incidence of projective planes of order q a prime power, are (q+1, 6)-cages. Also by using other incidence structures such as the generalized quadrangles or generalized hexagons, it can be obtained families of cages of girths 8 and 12. In this thesis, we present a construction of an entire family of girth 7 cages that arises from some combinatorial properties of the incidence graphs of generalized quadrangles of order (q,q)

Generation of Graph Classes with Efficient Isomorph Rejection

In this thesis, efficient isomorph-free generation of graph classes with the method of generation by canonical construction path(GCCP) is discussed. The method GCCP has been invented by McKay in the 1980s. It is a general method to recursively generate combinatorial objects avoiding isomorphic copies. In the introduction chapter, the method of GCCP is discussed and is compared to other well-known methods of generation. The generation of the class of quartic graphs is used as an example to explain this method. Quartic graphs are simple regular graphs of degree four. The programs, we developed based on GCCP, generate quartic graphs with 18 vertices more than two times as efficiently as the well-known software GENREG does. This thesis also demonstrates how the class of principal graph pairs can be generated exhaustively in an efficient way using the method of GCCP. The definition and importance of principal graph pairs come from the theory of subfactors where each subfactor can be modelled as a principal graph pair. The theory of subfactors has applications in the theory of von Neumann algebras, operator algebras, quantum algebras and Knot theory as well as in design of quantum computers. While it was initially expected that the classification at index 3 + √5 would be very complicated, using GCCP to exhaustively generate principal graph pairs was critical in completing the classification of small index subfactors to index 5¼. The other set of classes of graphs considered in this thesis contains graphs without a given set of cycles. For a given set of graphs, H, the Turán Number of H, ex(n,H), is defined to be the maximum number of edges in a graph on n vertices without a subgraph isomorphic to any graph in H. Denote by EX(n,H), the set of all extremal graphs with respect to n and H, i.e., graphs with n vertices, ex(n,H) edges and no subgraph isomorphic to any graph in H. We consider this problem when H is a set of cycles. New results for ex(n, C) and EX(n, C) are introduced using a set of algorithms based on the method of GCCP. Let K be an arbitrary subset of {C3, C4, C5, . . . , C32}. For given n and a set of cycles, C, these algorithms can be used to calculate ex(n, C) and extremal graphs in Ex(n, C) by recursively extending smaller graphs without any cycle in C where C = K or C = {C3, C5, C7, . . .} ᴜ K and n≤64. These results are considerably in excess of the previous results of the many researchers who worked on similar problems. In the last chapter, a new class of canonical relabellings for graphs, hierarchical canonical labelling, is introduced in which if the vertices of a graph, G, is canonically labelled by {1, . . . , n}, then G\{n} is also canonically labelled. An efficient hierarchical canonical labelling is presented and the application of this labelling in generation of combinatorial objects is discussed