141,313 research outputs found
ON STAR COLORING OF DEGREE SPLITTING OF COMB PRODUCT GRAPHS
A star coloring of a graph G is a proper vertex coloring in which every path on four vertices in G is not bicolored. The star chromatic number χs (G) of G is the least number of colors needed to star color G. Let G = (V,E) be a graph with V = S1 [ S2 [ S3 [ . . . [ St [ T where each Si is a set of all vertices of the same degree with at least two elements and T =V (G) − St i=1 Si. The degree splitting graph DS (G) is obtained by adding vertices w1,w2, . . .wt and joining wi to each vertex of Si for 1 i t. The comb product between two graphs G and H, denoted by G ⊲ H, is a graph obtained by taking one copy of G and |V (G)| copies of H and grafting the ith copy of H at the vertex o to the ith vertex of G. In this paper, we give the exact value of star chromatic number of degree splitting of comb product of complete graph with complete graph, complete graph with path, complete graph with cycle, complete graph with star graph, cycle with complete graph, path with complete graph and cycle with path graph
Detour Global Domination for Degree Splitting graphs of some graphs
In this paper, we introduced the new concept detour global domination number for degree splitting graph of standard graphs. The detour global dominating sets in some standard and special graphs are determined. First we recollect the concept of degree splitting graph of a graph and we produce some results based on the detour global domination number of degree splitting graph of star graph, bistar graph, complete bipartite graph, complete graph path graph, cycle graph, wheel graph and helm graph. A set S is called a detour global dominating set of G if S is both detour and global dominating set of G. The detour global domination number is the minimum cardinality of a detour global dominating set in G
Single-Strip Triangulation of Manifolds with Arbitrary Topology
Triangle strips have been widely used for efficient rendering. It is
NP-complete to test whether a given triangulated model can be represented as a
single triangle strip, so many heuristics have been proposed to partition
models into few long strips. In this paper, we present a new algorithm for
creating a single triangle loop or strip from a triangulated model. Our method
applies a dual graph matching algorithm to partition the mesh into cycles, and
then merges pairs of cycles by splitting adjacent triangles when necessary. New
vertices are introduced at midpoints of edges and the new triangles thus formed
are coplanar with their parent triangles, hence the visual fidelity of the
geometry is not changed. We prove that the increase in the number of triangles
due to this splitting is 50% in the worst case, however for all models we
tested the increase was less than 2%. We also prove tight bounds on the number
of triangles needed for a single-strip representation of a model with holes on
its boundary. Our strips can be used not only for efficient rendering, but also
for other applications including the generation of space filling curves on a
manifold of any arbitrary topology.Comment: 12 pages, 10 figures. To appear at Eurographics 200
The edge slide graph of the n-dimensional cube : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatū, New Zealand
The goal of this thesis is to understand the spanning trees of the n-dimensional cube Qn by
understanding their edge slide graph. An edge slide is a move that “slides” an edge of a spanning
tree of Qn across a two-dimensional face, and the edge slide graph is the graph on the spanning
trees of Qn with an edge between two trees if they are connected by an edge slide. Edge slides
are a restricted form of an edge move, in which the edges involved in the move are constrained
by the structure of Qn, and the edge slide graph is a subgraph of the tree graph of Qn given by
edge moves.
The signature of a spanning tree of Qn is the n-tuple (a1; : : : ; an), where ai is the number of
edges in the ith direction. The signature of a tree is invariant under edge slides and is therefore
constant on connected components. We say that a signature is connected if the trees with that
signature lie in a single connected component, and disconnected otherwise. The goal of this
research is to determine which signatures are connected.
Signatures can be naturally classified as reducible or irreducible, with the reducible signatures
being further divided into strictly reducible and quasi-irreducible signatures. We determine
necessary and sufficient conditions for (a1; : : : ; an) to be a signature of Qn, and show that
strictly reducible signatures are disconnected. We conjecture that strict reducibility is the only
obstruction to connectivity, and present substantial partial progress towards an inductive proof
of this conjecture. In particular, we reduce the inductive step to the problem of proving under
the inductive hypothesis that every irreducible signature has a “splitting signature” for which
the upright trees with that signature and splitting signature all lie in the same component. We
establish this step for certain classes of signatures, but at present are unable to complete it for
all.
Hall’s Theorem plays an important role throughout the work, both in characterising the
signatures, and in proving the existence of certain trees used in the arguments
Mixing of the Averaging process and its discrete dual on finite-dimensional geometries
We analyze the -mixing of a generalization of the Averaging process
introduced by Aldous. The process takes place on a growing sequence of graphs
which we assume to be finite-dimensional, in the sense that the random walk on
those geometries satisfies a family of Nash inequalities. As a byproduct of our
analysis, we provide a complete picture of the total variation mixing of a
discrete dual of the Averaging process, which we call Binomial Splitting
process. A single particle of this process is essentially the random walk on
the underlying graph. When several particles evolve together, they interact by
synchronizing their jumps when placed on neighboring sites. We show that, given
the number of particles and the (growing) size of the underlying graph,
the system exhibits cutoff in total variation if and .
Finally, we exploit the duality between the two processes to show that the
Binomial Splitting satisfies a version of Aldous' spectral gap identity,
namely, the relaxation time of the process is independent of the number of
particles.Comment: 30 pages. Typos fixe
The Complexity of Cluster Vertex Splitting and Company
Clustering a graph when the clusters can overlap can be seen from three
different angles: We may look for cliques that cover the edges of the graph, we
may look to add or delete few edges to uncover the cluster structure, or we may
split vertices to separate the clusters from each other. Splitting a vertex
means to remove it and to add two new copies of and to make each previous
neighbor of adjacent with at least one of the copies. In this work, we
study the underlying computational problems regarding the three angles to
overlapping clusterings, in particular when the overlap is small. We show that
the above-mentioned covering problem, which also has been independently studied
in different contexts,is NP-complete. Based on a previous so-called
critical-clique lemma, we leverage our hardness result to show that Cluster
Editing with Vertex Splitting is also NP-complete, resolving an open question
by Abu-Khzam et al. [ISCO 2018]. We notice, however, that the proof of the
critical-clique lemma is flawed and we give a counterexample. Our hardness
result also holds under a version of the critical-clique lemma to which we
currently do not have a counterexample. On the positive side, we show that
Cluster Vertex Splitting admits a vertex-linear problem kernel with respect to
the number of splits.Comment: 30 pages, 9 figure
): Pg.282-287 JOURNAL OF COMPUTER AND MATHEMATICAL SCIENCES An International Open Free Access
ABSTRACT The packing chromatic number of a graph G is the smallest integer k for which there exists a mapping such that any two vertices of color i are at distance at least 1 i + . It is a frequency assignment problem used in wireless networks, which is also called broadcast coloring. It is proved that packing coloring is NP-complete for general graphs and even for trees. In this paper, we give the packing chromatic number for splitting of bi star graph, sierpiński graph, broken wheel, jahangir graph and 4 q P K
Complete-Graph Tensor Network States: A New Fermionic Wave Function Ansatz for Molecules
We present a new class of tensor network states that are specifically
designed to capture the electron correlation of a molecule of arbitrary
structure. In this ansatz, the electronic wave function is represented by a
Complete-Graph Tensor Network (CGTN) ansatz which implements an efficient
reduction of the number of variational parameters by breaking down the
complexity of the high-dimensional coefficient tensor of a
full-configuration-interaction (FCI) wave function. We demonstrate that CGTN
states approximate ground states of molecules accurately by comparison of the
CGTN and FCI expansion coefficients. The CGTN parametrization is not biased
towards any reference configuration in contrast to many standard quantum
chemical methods. This feature allows one to obtain accurate relative energies
between CGTN states which is central to molecular physics and chemistry. We
discuss the implications for quantum chemistry and focus on the spin-state
problem. Our CGTN approach is applied to the energy splitting of states of
different spin for methylene and the strongly correlated ozone molecule at a
transition state structure. The parameters of the tensor network ansatz are
variationally optimized by means of a parallel-tempering Monte Carlo algorithm
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