Hubungan Dimensi Metrik Ketetanggaan dan Dimensi Metrik Ketetanggan Lokal Graf Hasil Operasi Kali Korona

Adjacency metric dimension and local adjacency metric dimension are the development of metric dimension. The purpose of this research is to determine the adjacency metric dimension of corona graph between any connected graph G and non-trivial graph H denoted by dimA(G⊙H), to determine the local adjacency metric dimension of corona graph between any connected graph G and non-trivial graph H denoted by dimA,l(G⊙H), and to determine the correlation between adjacency metric dimension and local adjacency metric dimension of corona product graph operations. In this research, it is found out that the value of adjacency metric dimension of G⊙H graph is affected by the basic characteristic of H and the domination characteristic. Meanwhile, the value of local adjacency metric dimension of G⊙H graph is only affected by the basic characteristic of H Futhermore, it is found a correlation of adjacency metric dimension and local adjacency metric dimension of corona product graph between any connected graph G and non-trivial graph H.

Massive Parallelization of the Global Hydrological Model mHM

<p>Parameter estimation of a global-scale, high-resolution hydrological model requires a powerful supercomputer and an optimized parallelization<br>algorithm. Improving the efficiency of such an implementation is essential to advance hydrological science and to minimize the uncertainty of<br>the major hydrologic fluxes and storages at continental and global scales. Within the ESM project [1], the main transfer-function parameters of the mHM<br>model will be estimated by jointly assimilating evapotranspiration (ET) from FLUXNET, the TWS anomaly from GRACE (NASA) and streamflow time series<br>from 5500 GRDC gauges to achieve this goal.</p><p>For the parallelization of the objective functions, a hybrid MPI-OpenMP scheme is implemented. While the parallelization<br>into equally sized subdomains for cell-wise computations  of fluxes (e.g., ET, TWS) is trivial,<br>cell-to-cell fluxes need to be computed for streamflow routing. For time series<br>datasets, the advanced parallelization algorithm MPI parallelized Decomposition of Forest (MDF) will be used. </p><p>In this study, we go beyond the standard approach which decomposes the river into tributaries (e.g. the Pfaffenstetter System<br>[2]). We apply a non-trivial graph algorithm to decompose each river-network into a tree data structure with nodes representing<br>subbasin domains of almost equal size [3]. </p><p>We analyze several aspects affecting the MDF parallelization: <br>(1) the communication time between nodes; (2) buffering data before sending; (3) optimizing total node idle time and total run time; (4) memory<br>imbalance between master processes and other processes. </p><p>We run the mHM model on the high-performance JUWELS supercomputer at Jülich Supercomputing Center (JSC) where the (routing) code efficiently scales up to ~180 nodes with 96 CPUs each. We discuss different parallelization aspects, <br>including the effect of parameters onto the scaling of MDF and we show the benefits of MDF over a non-parallelized routing module.</p><p>[1] https://www.esm-project.net/<br>[2] http://proceedings.esri.com/library/userconf/proc01/professional/papers/pap1008/p1008.htm<br>[3] https://meetingorganizer.copernicus.org/EGU2019/EGU2019-8129-1.pdf</p>

Spectral Dynamics of Graph Sequences Generated by Subdivision and Triangle Extension

For a graph G and a unary graph operation X, there is a graph sequence \G_k generated by G_0=G and G_{k+1}=X(G_k). Let Sp({G_k}) denote the set of normalized Laplacian eigenvalues of G_k. The set of limit points of \bigcup_{k=0}^\infty Sp(G_k)$, $\liminf_{k\rightarrow\infty}Sp(G_k) and $\limsup_{k\rightarrow \infty}Sp(G_k)$ are considered in this paper for graph sequences generated by two operations: subdivision and triangle extension. It is obtained that the spectral dynamic of graph sequence generated by subdivision is determined by a quadratic function, which is closely related to the the well-known logistic map; while that generated by triangle extension is determined by a linear function. By using the knowledge of dynamic system, the spectral dynamics of graph sequences generated by these two operations are characterized. For example, it is found that, for any initial non-trivial graph $G$, chaos takes place in the spectral dynamics of iterated subdivision graphs, and the set of limit points is the entire closed interval [0,2].

The Average Lower Connectivity of Graphs

For a vertexvof a graphG, thelower connectivity, denoted bysv(G), is the smallest number of vertices that containsvand those vertices whose deletion fromGproduces a disconnected or a trivial graph. The average lower connectivity denoted byκav(G)is the value(∑v∈VGsvG)/VG. It is shown that this parameter can be used to measure the vulnerability of networks. This paper contains results on bounds for the average lower connectivity and obtains the average lower connectivity of some graphs.

On Graphs with Cyclic Defect or Excess

The Moore bound constitutes both an upper bound on the order of a graph of maximum degree $d$ and diameter $D=k$ and a lower bound on the order of a graph of minimum degree $d$ and odd girth $g=2k+1$. Graphs missing or exceeding the Moore bound by $\epsilon$ are called graphs with defect or excess $\epsilon$, respectively. While Moore graphs (graphs with $\epsilon=0$) and graphs with defect or excess 1 have been characterized almost completely, graphs with defect or excess 2 represent a wide unexplored area. Graphs with defect (excess) 2 satisfy the equation $G_{d,k}(A) = J_n + B$ ($G_{d,k}(A) = J_n - B$), where $A$ denotes the adjacency matrix of the graph in question, $n$ its order, $J_n$ the $n\times n$ matrix whose entries are all 1's, $B$ the adjacency matrix of a union of vertex-disjoint cycles, and $G_{d,k}(x)$ a polynomial with integer coefficients such that the matrix $G_{d,k}(A)$ gives the number of paths of length at most $k$ joining each pair of vertices in the graph. In particular, if $B$ is the adjacency matrix of a cycle of order $n$ we call the corresponding graphs graphs with cyclic defect or excess; these graphs are the subject of our attention in this paper. We prove the non-existence of infinitely many such graphs. As the highlight of the paper we provide the asymptotic upper bound of $O(\frac{64}3d^{3/2})$ for the number of graphs of odd degree $d\ge3$ and cyclic defect or excess. This bound is in fact quite generous, and as a way of illustration, we show the non-existence of some families of graphs of odd degree $d\ge3$ and cyclic defect or excess. Actually, we conjecture that, apart from the Möbius ladder on 8 vertices, no non-trivial graph of any degree $\ge 3$ and cyclic defect or excess exists.

EFFICIENT ALGORITHM FOR OPTIMAL MATRIX ORTHOGONAL DECOMPOSITION PROBLEM IN INTENSITY-MODULATED RADIATION THERAPY

In this paper, we study an interesting matrix decomposition problem that seeks to decompose a "complicated" matrix into two "simpler" matrices while minimizing the sum of the horizontal complexity of the first sub-matrix and the vertical complexity of the second sub-matrix. The matrix decomposition problem is crucial for improving the "step-and-shoot" delivery efficiency in Intensity-Modulated Radiation Therapy, which aims to deliver a highly conformal radiation dose to a target tumor while sparing the surrounding normal tissues. Our algorithm is based on a non-trivial graph construction scheme, which enables us to formulate the decomposition problem as computing a minimum s-t cut in a 3-D geometric multi-pillar graph. Experiments on randomly generated intensity map matrices and on clinical data demonstrated the efficacy of our algorithm.

Distinguishing Numbers for Graphs and Groups

A graph $G$ is distinguished if its vertices are labelled by a map $\phi: V(G) \longrightarrow \{1,2,\ldots, k\}$ so that no non-trivial graph automorphism preserves $\phi$. The distinguishing number of $G$ is the minimum number $k$ necessary for $\phi$ to distinguish the graph. It measures the symmetry of the graph. We extend these definitions to an arbitrary group action of $\Gamma$ on a set $X$. A labelling $\phi: X \longrightarrow \{1,2,\ldots,k\}$ is distinguishing if no element of $\Gamma$ preserves $\phi$ except those which fix each element of $X$. The distinguishing number of the group action on $X$ is the minimum $k$ needed for $\phi$ to distinguish the group action. We show that distinguishing group actions is a more general problem than distinguishing graphs. We completely characterize actions of $S_n$ on a set with distinguishing number $n$, answering an open question of Albertson and Collins.