Mathematical Research Seminar - Archive

Let $G=(V,E)$ be a finite undirected graph of order $n$ and of size $m$. Let $\Delta$ and $\delta$ be the largest and the smallest degree of $G$, respectively. The spectral radius of $G$ is the largest eigenvalue of the adjacency
matrix of the graph $G$.
In this talk new bounds on the spectral radius of $\{C_3,C_4\}$-free graphs in terms of $m, n, \Delta$ and~$\delta$ will be presented.
Computer search shows that in most of the cases the bounds derived in this research are better than the existing bounds.
Joint work with Dragan Stevanović.

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Isogeometric analysis was first introduced as an approach for solving partial differential equations (PDE) that aims towards integration of worlds of Computer aided design (CAD) and Finite element analysis (FEA) by employing the same spaces of functions in domain representation as well as describing the solution of the PDE on that domain. In comparison to traditional finite element functions, it has been observed that functions of higher smoothness have positive effect on stability and convergence properties of numerical solution. Using isogeometric analysis for solving PDEs on multi-patch spline surfaces is an active area of research. But not every multi-patch spline surface is suitable for such task. Analysis-suitable $G^1$ (AS-$G^1$) multi-patch spline surfaces [1, 2] are particular $G^1$-smooth multi-patch spline surfaces, which are needed to ensure the construction of $C^1$-smooth multi-patch spline spaces with optimal polynomial reproduction properties. In [3], a global method to construct AS-$G^1$ planar multi-patch parameterisations has been developed. In this talk we present a locally based approach for the design of AS-$G^1$ multi-patch spline surfaces. The approach is based on a Lagrange multiplier method and generates AS-$G1$ multi-patch spline surfaces by approximating a given $G^1$-smooth but non-AS-$G^1$ multi- patch surface. Several numerical examples demonstrate the potential of the presented technique for the construction of AS-$G^1$ multi-patch spline surfaces and show that these surfaces are especially suited for applications in isogeometric analysis by solving the biharmonic problem, a particular fourth order partial differential equation, over them.

Joint work with: A. Farahat, M. Kapl, V. Vitrih.

References:

[1] A. Collin, G. Sangalli, and T. Takacs: Analysis-suitable G1 multi-patch parametriza- tions for C1 isogeometric spaces, Comput. Aided Geom. Des., 47 (2016), 93–113.

[2] A. Farahat, B. Jüttler, M. Kapl, and T. Takacs: Isogeometric analysis with C1- smooth functions over multi-patch surfaces, Comput. Methods Appl. Mech. Engrg. 403 (2023), 115706.

[3] M. Kapl, G. Sangalli, and T. Takacs: Construction of analysis-suitable G1 planar multi-patch parameterizations, Comput. Aided Des., 97 (2018), 41–55.

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In a connected graph G; a set of vertices S \subseteq V (G) locates all vertices of G if forevery pair of vertices u; v from G there exists at least one vertex s in S such that the distances from u; v to s are distinct. The cardinality of the smallest set of vertices that locates every element of V (G) is the (vertex) metric dimension of G. The motivation for such notion is the problem of finding the smallest possible number of sensors and the locations in a network for them to be instaled, so that every object in a netvork can be located by measuring distances to the sensors. Similarly as with locating vertices, the problem of locating edges is defined and the corresponding edge metric dimension of G. We start with considering metric dimensions of unicyclic graphs, where we establish an upper and lower bound of metric dimensions of such graphs, which imply that they obtain values from two particular consecutive integers. The condition under which the dimensions take each of the two possible values is then established, this condition involves three graph configurations per each dimension. This approach then naturally extends to cacti, i.e. graphs
with edge disjoint cycle. The exact value of metric dimensions for cacti yields a simple upper bound on the dimensions of cacti, and we conjecture that this bound holds also for all connected graphs. We give several results on graphs with minimum degree at least two and 2-connected graphs which support such conjecture. Joint work with Riste Škrekovski.

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Two matrices A,B are called similar if there is an invertible matrix P satisfying AP=PB. As is well known, complex matrices are up to similarity uniquely determined by their Jordan canonical form. This talk will discuss possible extensions to (joint) similarity of tuples of matrices. Tuples (A_1,...,A_n) and (B_1,...,B_n) are called similar if there is an invertible matrix P such that A_jP=PB_j for all j. The classification of matrix tuples up to similarity has been deemed a “hopeless problem”, but is widely studied due to its importance in multiple areas of mathematics, ranging from operator theory, invariant and representation theory and algebraic geometry to algebraic statistics and computational complexity. In this talk we shall present a new natural collection of separating invariants for matrix tuples, along the way solving a 2003 conjecture of Hadwin and Larson.

This is based on joint work with Harm Derksen, Visu Makam and Jurij Volčič.

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