303 research outputs found
Extremal trees, unicyclic and bicyclic graphs with respect to -Sombor spectral radii
For a graph and , denote by (or for
short) the degree of vertex . The -Sombor matrix
() of a graph is a square matrix,
where the -entry is equal to if the vertices and are
adjacent, and 0 otherwise. The -Sombor spectral radius of , denoted by
, is the largest eigenvalue of
the -Sombor matrix . In this paper, we consider
the extremal trees, unicyclic and bicyclic graphs with respect to the
-Sombor spectral radii. We characterize completely the extremal graphs with
the first three maximum Sombor spectral radii, which answers partially a
problem posed by Liu et al. in [MATCH Commun. Math. Comput. Chem. 87 (2022)
59-87]
The bounds of vertex Padmakar-Ivan index on k-trees
© 2019 by the authors. The Padmakar-Ivan (PI) index is a distance-based topological index and a molecular structure descriptor, which is the sum of the number of vertices over all edges uv of a graph such that these vertices are not equidistant from u and v. In this paper, we explore the results of PI-indices from trees to recursively clustered trees, the k-trees. Exact sharp upper bounds of PI indices on k-trees are obtained by the recursive relationships, and the corresponding extremal graphs are given. In addition, we determine the PI-values on some classes of k-trees and compare them, and our results extend and enrich some known conclusions
Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations
One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology
Detecting Topological Entanglement Entropy in a Lattice of Quantum Harmonic Oscillators
The Kitaev surface-code model is the most studied example of a topologically
ordered phase and typically involves four-spin interactions on a
two-dimensional surface. A universal signature of this phase is topological
entanglement entropy (TEE), but due to low signal to noise, it is extremely
difficult to observe in these systems, and one usually resorts to measuring
anyonic statistics of excitations or non-local string operators to reveal the
order. We describe a continuous-variable analog to the surface code using
quantum harmonic oscillators on a two-dimensional lattice, which has the
distinctive property of needing only two-body nearest-neighbor interactions for
its creation. Though such a model is gapless, satisfies an area law, and the
ground state can be simply prepared by measurements on a finitely squeezed and
gapped two-dimensional cluster state, which does not have topological order.
Asymptotically, the TEE grows linearly with the squeezing parameter, and we
show that its mixed-state generalization, the topological mutual information,
is robust to some forms of state preparation error and can be detected simply
using single-mode quadrature measurements. Finally, we discuss scalable
implementation of these methods using optical and circuit-QED technology.Comment: 16 pages, 7 figures, added section about correlations length and
study of the topological logarithmic negativity. Typos fixed. Comments
welcom
Symmetry in Graph Theory
This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view
Cliques, colouring and satisfiability : from structure to algorithms
We examine the implications of various structural restrictions on the computational
complexity of three central problems of theoretical computer science
(colourability, independent set and satisfiability), and their relatives. All problems
we study are generally NP-hard and they remain NP-hard under various restrictions.
Finding the greatest possible restrictions under which a problem is computationally
difficult is important for a number of reasons. Firstly, this can make it easier to
establish the NP-hardness of new problems by allowing easier transformations. Secondly,
this can help clarify the boundary between tractable and intractable instances
of the problem.
Typically an NP-hard graph problem admits an infinite sequence of narrowing
families of graphs for which the problem remains NP-hard. We obtain a number
of such results; each of these implies necessary conditions for polynomial-time
solvability of the respective problem in restricted graph classes. We also identify
a number of classes for which these conditions are sufficient and describe explicit
algorithms that solve the problem in polynomial time in those classes. For the
satisfiability problem we use the language of graph theory to discover the very first
boundary property, i.e. a property that separates tractable and intractable instances
of the problem. Whether this property is unique remains a big open problem
Topics in graph colouring and extremal graph theory
In this thesis we consider three problems related to colourings of graphs and one problem in extremal graph theory. Let be a connected graph with vertices and maximum degree . Let denote the graph with vertex set all proper -colourings of and two -colourings are joined by an edge if they differ on the colour of exactly one vertex.
Our first main result states that has a unique non-trivial component with diameter . This result can be viewed as a reconfigurations analogue of Brooks' Theorem and completes the study of reconfigurations of colourings of graphs with bounded maximum degree.
A Kempe change is the operation of swapping some colours , of a component of the subgraph induced by vertices with colour or . Two colourings are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. Our second main result states that all -colourings of a graph are Kempe equivalent unless is the complete graph or the triangular prism. This settles a conjecture of Mohar (2007).
Motivated by finding an algorithmic version of a structure theorem for bull-free graphs due to Chudnovsky (2012), we consider the computational complexity of deciding if the vertices of a graph can be partitioned into two parts such that one part is triangle-free and the other part is a collection of complete graphs. We show that this problem is NP-complete when restricted to five classes of graphs (including bull-free graphs) while polynomial-time solvable for the class of cographs.
Finally we consider a graph-theoretic version formulated by Holroyd, Spencer and Talbot (2007) of the famous Erd\H{o}s-Ko-Rado Theorem in extremal combinatorics and obtain some results for the class of trees
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