N.D. Verhulst
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Graph burning is a discrete-time process that models the spread of social contagion. Initially, all vertices are unburned. In each round, one unburned vertex is selected and burned, while any unburned vertex that has a burned neighbour from the previous round also becomes burned. The burning number of a graph is the minimum number of rounds needed to burn the entire graph. In this paper, we study the burning number of graph powers. First, we show that for a connected graph G, its graph power Gk contains a (k + 1)+-branching tree as a spanning tree. A (k + 1)+-branching tree is one in which all internal vertices have degree at least k + 1. We then show that (k + 1)+-branching trees on n vertices have burning number at most [√4(k-1)n/k2] . As the burning number of a graph is at most the burning number of any of its spanning trees, this gives an upper bound on the burning number of graph powers. We also derive an alternative upper bound on the burning number of k+-branching trees using the strongest currently known general burning number bound [Bastide et al.]. We then identify the ranges of k and n for which our_ bound outperforms or matches this alternative bound. Finally, we show that b(Gk) ≤ (1 + o(1))√n/k based on the asymptotic burning number bound of Norin and Turcotte.
The main concepts of linear algebra are introduced from a geometrical perspective. We start by introducing the basic concepts of vectors, lines, and planes. There follows a thorough treatment of standard subjects like systems of linear equations, matrix arithmetic, eigenvalues and eigenvectors, orthogonality etc. In the final chapters, more advanced topics like symmetric matrices and discrete dynamical systems are discussed.
Throughout the book, many interactive applets are inserted to give the student hands-on experience with linear algebra. Thanks to an ample selection of embedded exercises with individualized feedback, the book offers a stimulating learning environment for studying linear algebra!
This open interactive textbook was developed by a team of lecturers and developers from the Delft Institute of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Science, from the TU Delft University of Technology. ...
The main concepts of linear algebra are introduced from a geometrical perspective. We start by introducing the basic concepts of vectors, lines, and planes. There follows a thorough treatment of standard subjects like systems of linear equations, matrix arithmetic, eigenvalues and eigenvectors, orthogonality etc. In the final chapters, more advanced topics like symmetric matrices and discrete dynamical systems are discussed.
Throughout the book, many interactive applets are inserted to give the student hands-on experience with linear algebra. Thanks to an ample selection of embedded exercises with individualized feedback, the book offers a stimulating learning environment for studying linear algebra!
This open interactive textbook was developed by a team of lecturers and developers from the Delft Institute of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Science, from the TU Delft University of Technology.
The main concepts of linear algebra are introduced from a geometrical perspective. We start by introducing the basic concepts of vectors, lines, and planes. There follows a thorough treatment of standard subjects like systems of linear equations, matrix arithmetic, eigenvalues and eigenvectors, orthogonality etc. In the final chapters, more advanced topics like symmetric matrices and discrete dynamical systems are discussed.
Throughout the book, many interactive applets are inserted to give the student hands-on experience with linear algebra. Thanks to an ample selection of embedded exercises with individualized feedback, the book offers a stimulating learning environment for studying linear algebra! ...
The main concepts of linear algebra are introduced from a geometrical perspective. We start by introducing the basic concepts of vectors, lines, and planes. There follows a thorough treatment of standard subjects like systems of linear equations, matrix arithmetic, eigenvalues and eigenvectors, orthogonality etc. In the final chapters, more advanced topics like symmetric matrices and discrete dynamical systems are discussed.
Throughout the book, many interactive applets are inserted to give the student hands-on experience with linear algebra. Thanks to an ample selection of embedded exercises with individualized feedback, the book offers a stimulating learning environment for studying linear algebra!
When does the complex product of a given number of subsets of a group generate the same subgroup as their union? We answer this question in a more general form by introducing HS-stability and characterising the HS-stable involution subsemigroup generated by a subset of a given involution semigroup. We study HS-stability for the special cases of regular ∗-semigroups and commutative involution semigroups.