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Oktavianingtyas Ervin - One of the best experts on this subject based on the ideXlab platform.

  • ANALISIS DUA KONEKSI PELANGI PADA GRAF HASIL OPERASI PERKALIAN ARTESIAN GRAF KIPAS F_(1,3) DAN GRAF LINGKARAN C_n SERTA KAITANNYA DENGAN KETERAMPILAN BERPIKIR TINGKAT TINGGI
    'UPT Penerbitan Universitas Jember', 2018
    Co-Authors: Wijayanti Elsy, Dafik D, Oktavianingtyas Ervin
    Abstract:

    Abstract. The rainbow connection of the graph of G = (V, E) if for each pair of points u and v in G there is a path with points u and v as the end points on each side obtaining different colors, the path is called the rainbow path. The rainbow-connected number graph G is the Smallest Positive Integer such that G has a rainbow connection staining denoted rc(G). While the graph is said to be a rainbow 2-connected staining on G if for each pair of points u and v on the sides there are 2 different paths, with u and v being the end points of each side obtaining different colors and 2 passages can not intersect each other, denoted rc2 (G). Let G be the graph connected with  so that , with d is degrees. Let G be a rainbow   so that where κ is the number of paths of rainbow connecting every two distinct points in G. The results of this research are new theorem about the rainbow connection and 2-rainbow connected. Keywords : rainbow connection, 2-rainbow connected, rainbow pat

  • ANALISIS DUA KONEKSI PELANGI PADA GRAF HASIL OPERASI PERKALIAN KARTESIAN GRAF KIPAS F_(1,3) DAN GRAF LINGKARAN C_n SERTA KAITANNYA DENGAN KETERAMPILAN BERPIKIR TINGKAT TINGGI
    'UPT Penerbitan Universitas Jember', 2018
    Co-Authors: Wijayanti Elsy, Dafik Dafik, Oktavianingtyas Ervin
    Abstract:

    Abstract. The rainbow connection of the graph of G = (V, E) if for each pair of points u and v in G there is a path with points u and v as the end points on each side obtaining different colors, the path is called the rainbow path. The rainbow-connected number graph G is the Smallest Positive Integer such that G has a rainbow connection staining denoted rc(G). While the graph is said to be a rainbow 2-connected staining on G if for each pair of points u and v on the sides there are 2 different paths, with u and v being the end points of each side obtaining different colors and 2 passages can not intersect each other, denoted rc2 (G). Let G be the graph connected with  so that , with d is degrees. Let G be a rainbow   so that where κ is the number of paths of rainbow connecting every two distinct points in G. The results of this research are new theorem about the rainbow connection and 2-rainbow connected.   Keywords : rainbow connection, 2-rainbow connected, rainbow pat

Wijayanti Elsy - One of the best experts on this subject based on the ideXlab platform.

  • ANALISIS DUA KONEKSI PELANGI PADA GRAF HASIL OPERASI PERKALIAN ARTESIAN GRAF KIPAS F_(1,3) DAN GRAF LINGKARAN C_n SERTA KAITANNYA DENGAN KETERAMPILAN BERPIKIR TINGKAT TINGGI
    'UPT Penerbitan Universitas Jember', 2018
    Co-Authors: Wijayanti Elsy, Dafik D, Oktavianingtyas Ervin
    Abstract:

    Abstract. The rainbow connection of the graph of G = (V, E) if for each pair of points u and v in G there is a path with points u and v as the end points on each side obtaining different colors, the path is called the rainbow path. The rainbow-connected number graph G is the Smallest Positive Integer such that G has a rainbow connection staining denoted rc(G). While the graph is said to be a rainbow 2-connected staining on G if for each pair of points u and v on the sides there are 2 different paths, with u and v being the end points of each side obtaining different colors and 2 passages can not intersect each other, denoted rc2 (G). Let G be the graph connected with  so that , with d is degrees. Let G be a rainbow   so that where κ is the number of paths of rainbow connecting every two distinct points in G. The results of this research are new theorem about the rainbow connection and 2-rainbow connected. Keywords : rainbow connection, 2-rainbow connected, rainbow pat

  • ANALISIS DUA KONEKSI PELANGI PADA GRAF HASIL OPERASI PERKALIAN KARTESIAN GRAF KIPAS F_(1,3) DAN GRAF LINGKARAN C_n SERTA KAITANNYA DENGAN KETERAMPILAN BERPIKIR TINGKAT TINGGI
    'UPT Penerbitan Universitas Jember', 2018
    Co-Authors: Wijayanti Elsy, Dafik Dafik, Oktavianingtyas Ervin
    Abstract:

    Abstract. The rainbow connection of the graph of G = (V, E) if for each pair of points u and v in G there is a path with points u and v as the end points on each side obtaining different colors, the path is called the rainbow path. The rainbow-connected number graph G is the Smallest Positive Integer such that G has a rainbow connection staining denoted rc(G). While the graph is said to be a rainbow 2-connected staining on G if for each pair of points u and v on the sides there are 2 different paths, with u and v being the end points of each side obtaining different colors and 2 passages can not intersect each other, denoted rc2 (G). Let G be the graph connected with  so that , with d is degrees. Let G be a rainbow   so that where κ is the number of paths of rainbow connecting every two distinct points in G. The results of this research are new theorem about the rainbow connection and 2-rainbow connected.   Keywords : rainbow connection, 2-rainbow connected, rainbow pat

Mahmud Akelbek - One of the best experts on this subject based on the ideXlab platform.

  • a bound on the scrambling index of a primitive matrix using boolean rank
    Linear Algebra and its Applications, 2009
    Co-Authors: Mahmud Akelbek, Sandra Fital, Jian Shen
    Abstract:

    Abstract The scrambling index of an n × n primitive matrix A is the Smallest Positive Integer k such that A k ( A t ) k = J , where A t denotes the transpose of A and J denotes the n × n all ones matrix. For an m × n Boolean matrix M , its Boolean rank b ( M ) is the Smallest Positive Integer b such that M = AB for some m × b Boolean matrix A and b × n Boolean matrix B . In this paper, we give an upper bound on the scrambling index of an n × n primitive matrix M in terms of its Boolean rank b ( M ) . Furthermore we characterize all primitive matrices that achieve the upper bound.

  • a bound on the scrambling index of a primitive matrix using boolean rank
    arXiv: Combinatorics, 2009
    Co-Authors: Mahmud Akelbek, Sandra Fital, Jian Shen
    Abstract:

    The scrambling index of an $n\times n$ primitive matrix $A$ is the Smallest Positive Integer $k$ such that $A^k(A^{t})^k=J$, where $A^t$ denotes the transpose of $A$ and $J$ denotes the $n\times n$ all ones matrix. For an $m\times n$ Boolean matrix $M$, its {\it Boolean rank} $b(M)$ is the Smallest Positive Integer $b$ such that $M=AB$ for some $m \times b$ Boolean matrix $A$ and $b\times n$ Boolean matrix $B$. In this paper, we give an upper bound on the scrambling index of an $n\times n$ primitive matrix $M$ in terms of its Boolean rank $b(M)$. Furthermore we characterize all primitive matrices that achieve the upper bound.

  • coefficients of ergodicity and the scrambling index
    Linear Algebra and its Applications, 2009
    Co-Authors: Mahmud Akelbek, Steve Kirkland
    Abstract:

    For a primitive stochastic matrix S, upper bounds on the second largest modulus of an eigenvalue of S are very important, because they determine the asymptotic rate of convergence of the sequence of powers of the corresponding matrix. In this paper, we introduce the definition of the scrambling index for a primitive digraph. The scrambling index of a primitive digraph D is the Smallest Positive Integer k such that for every pair of vertices u and v, there is a vertex w such that we can get to w from u and v in D by directed walks of length k; it is denoted by k(D). We investigate the scrambling index for primitive digraphs, and give an upper bound on the scrambling index of a primitive digraph in terms of the order and the girth of the digraph. By doing so we provide an attainable upper bound on the second largest modulus of eigenvalues of a primitive matrix that make use of the scrambling index.

Khaochim Narissara - One of the best experts on this subject based on the ideXlab platform.

  • On the order of appearance of product of consecutive Fibonacci numbers
    'Elsevier BV', 2018
    Co-Authors: Pongsriiam Prapanpong, Khaochim Narissara
    Abstract:

    Let $F_{n}$ be the $n$th Fibonacci number. For each Positive Integer $m$, the order of appearance of $m$, denoted by $z(m)$, is the Smallest Positive Integer $k$ such that $m$ divides $F_k$. Recently, D. Marques has obtained a formula for $z(F_{n}F_{n+1})$, $z(F_{n}F_{n+1}F_{n+2})$, and $z(F_{n}F_{n+1}F_{n+2}F_{n+3})$. In this paper, we extend Marques' result to the case $z(F_{n}F_{n+1}\cdots F_{n+k})$, for every $4\leq k \leq 6$. For instance, we prove that, for $n\geq1$,\[z(F_{n}F_{n+1}F_{n+2}F_{n+3}F_{n+4}) =\begin{cases}a, & \text{if $n\equiv1,2,3,4,5,6,7,10 \pmod {12}$, or $n\equiv8,60 \pmod {72}$};\\2a, & \text{if $n\equiv9,11\pmod {12}$, or $n\equiv24,44 \pmod {72}$};\\3a, & \text{if $n\equiv12,32,36,56 \pmod {72}$}; \\6a, & \text{if $n\equiv0,20,48,68 \pmod {72}$}\end{cases}\] where $a=[n,n+1,n+2,n+3,n+4]$

Jayawardene C. J. - One of the best experts on this subject based on the ideXlab platform.

  • On Star critical Ramsey numbers related to large cycles versus complete graphs
    2020
    Co-Authors: Jayawardene C. J., Navaratna W. C. W.
    Abstract:

    Let $K_n$ denote the complete graph on $n$ vertices and $G, H$ be finite graphs. Consider a two-coloring of edges of $K_n$. When a copy of $G$ in the first color, red, or a copy of $H$ in the second color, blue is in $K_n$, we write $K_n\rightarrow (G,H)$. The Ramsey number $r(G, H)$ is defined as the Smallest Positive Integer $n$ such that $K_{n} \rightarrow (G, H)$. Star critical Ramsey $r_*(G, H)$ is defined as the largest value of $k$ such that $K_{r(G,H)-1} \sqcup K_{1,k} \rightarrow (G, H)$. In this paper, we find $r_*(C_n, K_m)$ for $m \geq 7$ and $n \geq (m-3)(m-1)$.Comment: 10 pages, 2 figures. arXiv admin note: text overlap with arXiv:1902.0264

  • On Star-critical (K1,n,K1,m + e) Ramsey numbers
    2020
    Co-Authors: Jayawardene C. J., Senadheera J. N., Fernando K. A. S. N., Navaratna W. C. W.
    Abstract:

    Let $G, H$ be finite graphs without loops or multiple edges and $K_n$ denote the complete graph on $n$ vertices. If for every red/blue colouring of edges of the complete graph $K_n$, there exists a red copy of $G$, or a blue copy of $H$, we will say that $K_n\rightarrow (G,H)$. The Ramsey number $r(G, H)$ is defined as the Smallest Positive Integer $n$ such that $K_{n} \rightarrow (G, H)$. Star-critical Ramsey number $r_*(G, H)$ is defined as the largest value of $k$ such that $K_{r(G,H)-1} \sqcup K_{1,k} \rightarrow (G, H)$. In this paper, we will find $r_*(K_{1,n}, K_{1,m}+e)$ for all $n,m \geq 3$.Comment: 8 pages, 5 figure

  • All Ramsey $(C_n,K_6)$ critical graphs for large $n$
    2020
    Co-Authors: Jayawardene C. J., Navaratna, Chandanie W. W., Senadheera J. N.
    Abstract:

    Let $G$ and $H$ be finite graphs. If for any two-coloring of the edges of a complete graph $K_n$, there is a copy of $G$ in the first color, red, or a copy of $H$ in the second color, blue, we will say $K_n\rightarrow (G,H)$. The Ramsey number $r(G, H)$ is defined as the Smallest Positive Integer $n$ such that $K_{n} \rightarrow (G, H)$. A two-coloring of $K_{r(G, H)-1}$ such that $K_{r(G, H)-1} \not \rightarrow (G,H)$ is called a critical coloring. A Ramsey critical $r(G, H)$ graph is a graph induced by the first color of a critical coloring. In this paper, when $n \geq 15$, we show that there exist exactly sixty eight non-isomorphic Ramsey critical $r(C_n, K_6)$ graphs.Comment: 14 pages, 4 figure