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114

Jul 25, 2018
07/18

by
I. N. (Ioannes N.) Grypares; Plato

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Book from Project Gutenberg: Πολιτεία, Τόμος 3

Topics: JC, Political science -- Early works to 1800, Utopias -- Early works to 1800, Justice -- Early...

Source: https://www.gutenberg.org/ebooks/39524

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78

Jul 25, 2018
07/18

by
Plato; I. N. (Ioannes N.) Grypares

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Book from Project Gutenberg: Πολιτεία, Τόμος 4

Topics: Utopias -- Early works to 1800, JC, Justice -- Early works to 1800, Political science -- Early...

Source: https://www.gutenberg.org/ebooks/39530

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83

Jul 26, 2018
07/18

by
I. N. (Ioannes N.) Grypares; Plato

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Book from Project Gutenberg: Πολιτεία, Τόμος 2

Topics: Utopias -- Early works to 1800, Justice -- Early works to 1800, Political science -- Early works to...

Source: https://www.gutenberg.org/ebooks/39493

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108

Jul 26, 2018
07/18

by
Plato; I. N. (Ioannes N.) Grypares

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Book from Project Gutenberg: Πολιτεία, Τόμος 1

Topics: Utopias -- Early works to 1800, Political science -- Early works to 1800, JC, Justice -- Early...

Source: https://www.gutenberg.org/ebooks/39476

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5.0

Jun 30, 2018
06/18

by
Johan Kok

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The concept of the brush number $b_r(G)$ was introduced for a simple connected undirected graph $G$. This note extends the concept to a special family of directed graphs and declares that the brush number $b_r(J_n(1))$ of a finite Jaco graph, $J_n(1), n \in \Bbb N$ with prime Jaconian vertex $v_i$ is given by:\\ \\ $b_r(J_n(1)) = \sum\limits_{j=1}^{I}(d^+(v_j) - d^-(v_j)) + \sum\limits_{j=I+1}^{n}max\{0, (n-j) - d^-(v_j)\}.

Topics: Mathematics, Combinatorics

Source: http://arxiv.org/abs/1412.5733

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3.0

Jun 30, 2018
06/18

by
Johan Kok

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Parking functions are well researched and interesting results are found in the listed references and more. Some introductory results stemming from application to degree sequences of simple connected graphs are provided in this paper. Amongst others, the result namely, that a derivative degree sequence, $d_d(G) \in \Bbb D_d(G)= \{(\lceil\frac{d(v_1}{\ell}\rceil, \lceil\frac{d(v_2)}{\ell}\rceil, \lceil\frac{d(v_3)}{\ell}\rceil, ..., \lceil\frac{d(v_n)}{\ell}\rceil| \ell = d(v_i), \forall i,$ with...

Topics: Mathematics, Combinatorics

Source: http://arxiv.org/abs/1409.4163

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6.0

Jun 30, 2018
06/18

by
Johan Kok

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Total irregularity of a simple undirected graph $G$ is defined to be $irr_t(G) = \frac{1}{2}\sum\limits_{u, v \in V(G)}|d(u) - d(v)|$. See Abdo and Dimitrov [2]. We allocate the \emph{Fibonacci weight,} $f_i$ to a vertex $v_j$ of a simple connected graph, if and only if $d(v_j) = i$ and define the \emph{total fibonaccian irregularity} or $f_t-irregularity$ denoted $firr_t(G)$ for brevity, as: $firr_t(G) = \sum\limits_{i=1}^{n-1}\sum\limits_{j=i+1}^{n}|f_i - f_j|.$ The concept of an...

Topics: Mathematics, Combinatorics

Source: http://arxiv.org/abs/1406.6168

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4.0

Jun 30, 2018
06/18

by
Johan Kok; Sudev Naduvath

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A vertex $v$ of a given graph is said to be in a rainbow neighbourhood of $G$ if every colour class of $G$ consists of at least one vertex from the closed neighbourhood $N[v]$. A maximal proper colouring of a graph $G$ is a Johan colouring if and only if every vertex of $G$ belongs to a rainbow neighbourhood of $G$. In general all graphs need not have a Johan colouring, even though they admit a chromatic colouring. In this paper, we characterise graphs which admit a Johan colouring. We also...

Topics: General Mathematics, Mathematics

Source: http://arxiv.org/abs/1704.02869

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5.0

Jun 28, 2018
06/18

by
Johan Kok; Naduvath Sudev

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The studies on $b$-chromatic number attracted much interest since its introduction. In this paper, we discuss the $b$-chromatic number of certain classes of graphs and digraphs. The notion of a new general family of graphs called the Chithra graphs of a graph $G$ are also introduced.

Topics: General Mathematics, Mathematics

Source: http://arxiv.org/abs/1511.00680

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11

Jun 29, 2018
06/18

by
Johan Kok; Naduvath Sudev

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Consider a network $D$ of pipes which have to be cleaned using some cleaning agents, called brushes, assigned to some vertices. The tattooing of a simple connected directed graph $D$ is a particular type of the cleaning in which an arc are coloured by the colour of the colour-brush transiting it and the tattoo number of $D$ is a corresponding derivative of brush numbers in it. In this paper, we introduce a new concept, called the tattoo index of a given graph $G$, which is an efficiency index...

Topics: General Mathematics, Mathematics

Source: http://arxiv.org/abs/1604.00519

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7.0

Jun 29, 2018
06/18

by
Johan Kok; Saptarshi Bej

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Recall that the minimum number of colors that allow a proper coloring of graph $G$ is called the chromatic number of $G$ and denoted by $\chi(G).$ In this paper the concepts of $\chi$'-chromatic sum and $\chi^+$-chromatic sum are introduced. The extended graph $G^x$ of a graph $G$ was recently introduced for certain regular graphs. We further the concepts of $\chi$'-chromatic sum and $\chi^+$-chromatic sum to extended paths and cycles. The paper concludes with \emph{patterned structured} graphs.

Topics: General Mathematics, Mathematics

Source: http://arxiv.org/abs/1602.03735

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12

Jun 29, 2018
06/18

by
Johan Kok; Naduvath Sudev

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Consider a network $D$ of pipes which have to be cleaned using some cleaning agents, called brushes, assigned to some vertices. The minimum number of brushes required for cleaning the network $D$ is called its brush number. The tattooing of a simple connected directed graph $D$ is a particular type of the cleaning in which an arc are coloured by the colour of the colour-brush transiting it and the tattoo number of $D$ is a corresponding derivative of brush numbers in it. Tattooing along an...

Topics: General Mathematics, Mathematics

Source: http://arxiv.org/abs/1603.00303

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6.0

Jun 30, 2018
06/18

by
Johan Kok; Sudev Naduvath

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The total irregularity of a simple undirected graph $G$ is denoted by $irr_t(G)$ and is defined as $irr_t(G) = \frac{1}{2}\sum\limits_{u,v \in V(G)}|d(u) - d(v)|$. In this paper, the concept called edge-transformation in relation to total irregularity of simple undirected graphs with at least one cut edge is introduced. We also introduce the concept of an edge-joint between two simple undirected graphs. We also introduce the concept of total irregularity in respect of in-degree and out-degree...

Topics: Mathematics, Combinatorics

Source: http://arxiv.org/abs/1406.6863

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4.0

Jun 30, 2018
06/18

by
Johan Kok; Vivian Mukungunugwa

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Kok et.al. [3] introduced Jaco Graphs \emph{(order 1)}. It is hoped that as a special case, a closed formula can be found for the number of edges of a finite Jaco Graph $J_n(1)$. However, the algorithms discussed in Ahlbach et.al. [1] suggest this might not be possible. Finding a closed formula for the number of edges of a Jaco Graph $J_n(1), n \in \Bbb N$ remains an interesting open problem. In this note we present three alternative, \emph{formula}.

Topics: Mathematics, Combinatorics

Source: http://arxiv.org/abs/1409.0656

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8.0

Jun 30, 2018
06/18

by
Johan Kok; Vivian Mukungunugwa

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The topological indices $irr(G)$ related to the \emph{first Zagreb index,} $M_1(G)$ and the \emph{second Zagreb index,} $M_2(G)$ are the oldest irregularity measures researched. Alberton $[3]$ introduced the \emph{irregularity} of $G$ as $irr(G) = \sum\limits_{e \in E(G)}imb(e), imb(e) = |d(v) - d(u)|_{e=vu}$. In the paper of Fath-Tabar $[7]$, Alberton's indice was named the \emph{third Zagreb indice} to conform with the terminology of chemical graph theory. Recently Ado et.al. $[1]$ introduced...

Topics: Mathematics, Combinatorics

Source: http://arxiv.org/abs/1407.8290

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3.0

Jun 30, 2018
06/18

by
Johan Kok; Susanth C

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Kok et.al. [7] introduced Jaco Graphs (\emph{order 1}). In this essay we present a recursive formula to determine the \emph{independence number} $\alpha(J_n(1)) = |\Bbb I|$ with, $\Bbb I = \{v_{i,j}| v_1 = v_{1,1} \in \Bbb I$ and $v_i = v_{i,j} =v_{(d^+(v_{m, (j-1)}) + m +1)}\}.$ We also prove that for the Jaco Graph, $J_n(1), n \in \Bbb N$ with the prime Jaconian vertex $v_i$ the chromatic number, $\chi(J_n(1))$ is given by: \begin{equation*} \chi(J_n(1)) \begin{cases} = (n-i) + 1,...

Topics: Mathematics, Combinatorics

Source: http://arxiv.org/abs/1410.8328

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9.0

Jun 30, 2018
06/18

by
Johan Kok; Susanth C

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The concept of the \emph{McPherson number} of a simple connected graph $G$ on $n$ vertices denoted by $\Upsilon(G)$, is introduced. The recursive concept, called the \emph{McPherson recursion}, is a series of \emph{vertex explosions} such that on the first interation a vertex $v \in V(G)$ explodes to arc (directed edges) to all vertices $u \in V(G)$ for which the edge $vu \notin E(G)$, to obtain the mixed graph $G'_1.$ Now $G'_1$ is considered on the second iteration and a vertex $w \in V(G'_1)...

Topics: Mathematics, Combinatorics

Source: http://arxiv.org/abs/1410.8637

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6.0

Jun 30, 2018
06/18

by
Johan Kok; Naduvath Sudev

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Sprout graphs are finite directed graphs matured over a finite subset of the non-negative time line. A simple undirected connected graph on at least two vertices is required to construct an infant graph to mature from. The maxi-max arc-weight principle and the maxi-min arc-weight principle are introduced in the paper. These principles are used to determine the maximum and minimum maturity weight of a sprout graph.

Topics: Mathematics, Combinatorics

Source: http://arxiv.org/abs/1403.8109

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13

Jun 27, 2018
06/18

by
Johan Kok; N. K. Sudev

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A hole of a simple connected graph $G$ is a chordless cycle $C_n,$ where $n \in \Bbb N, n \geq 4,$ in the graph $G$. The girth of a simple connected graph $G$ is the smallest cycle in $G$, if any such cycle exists. It can be observed that all such smallest cycles are necessarily chordless. We call the cycle $C_3$ in a given graph $G$ a primitive hole of that graph. We introduce the notion of the primitive hole number of a graph as the number of primitive holes present in that graph. In this...

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1503.04526

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7.0

Jun 30, 2018
06/18

by
Johan Kok; Naduvath Sudev; Muhammad Kamran Jamil

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In this paper, we introduce the notion of the rainbow neighbourhood and a related graph parameter namely, the rainbow neighbourhood number of a graph $G$. We report on preliminary results thereof. We also establish a necessary and sufficient condition for the existence of a rainbow neighbourhood in the line graph of a graph $G$.

Topics: General Mathematics, Mathematics

Source: http://arxiv.org/abs/1703.01089

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6.0

Jun 30, 2018
06/18

by
Johan Kok; Naduvath Sudev; Muhammad Kamran Jamil

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For a colour cluster $\C =(\mathcal{C}_1,\mathcal{C}_2, \mathcal{C}_3,\dots,\mathcal{C}_\ell)$, $\mathcal{C}_i$ is a colour class, and $|\mathcal{C}_i|=r_i \geq 1$, we investigate a simple connected graph structure $G^{\C}$, which represents a graphical embodiment of the colour cluster such that the chromatic number $\chi(G^{\C})= \ell,$ and the number of edges is a maximum, denoted $\varepsilon^+(G^{\C})$. We also extend the study by inducing new colour clusters recursively by blending the...

Topics: General Mathematics, Mathematics

Source: http://arxiv.org/abs/1702.00103

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7.0

Jun 29, 2018
06/18

by
Johan Kok; Naduvath Sudev; Muhammad Kamran Jamil

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For a colour cluster $\mathbb{C} =(\mathcal{C}_1,\mathcal{C}_2, \mathcal{C}_3,\ldots,\mathcal{C}_\ell)$, where $\mathcal{C}_i$ is a colour class such that $|\mathcal{C}_i|=r_i$, a positive integer, we investigate two types of simple connected graph structures $G^{\mathbb{C}}_1$, $G^{\mathbb{C}}_2$ which represent graphical embodiments of the colour cluster such that the chromatic numbers $\chi(G^{\mathbb{C}}_1)=\chi(G^{\mathbb{C}}_2)=\ell$ and...

Topics: General Mathematics, Mathematics

Source: http://arxiv.org/abs/1611.01416

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12

Jun 29, 2018
06/18

by
Johan Kok; N. K. Sudev; U. Mary

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In this paper we introduce a variation of the well-known Zagreb indices by considering a proper vertex colouring of a graph $G$. The chromatic Zagreb indices are defined in terms of the parameter $c(v), v \in V(G)$ instead of the invariant $d_G(v)$. The notion of chromatically stable graphs is also introduced.

Topics: General Mathematics, Mathematics

Source: http://arxiv.org/abs/1605.01350

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8.0

Jun 30, 2018
06/18

by
Johan Kok; Sudev Naduvath; Vivian Mukungunugwa

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In this paper, we introduce the notion of a finite non-simple directed graph, called an ornated graph and initiate a study on ornated graphs. An ornated graph is a directed graph on $n$ vertices, denoted by $O_n(s_l)$, whose vertices are consecutively labeled clockwise on the circumference of a circle and constructed from an ordered string $s_l$ joining them in such a way that for an odd indexed entry $a_t$ of the string, a tail $v_i$ has clockwise heads $v_j$ if and only if $(i+a_t) \ge j$ and...

Topics: Mathematics, Combinatorics

Source: http://arxiv.org/abs/1403.7916

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9.0

Jun 29, 2018
06/18

by
Johan Kok; Erika Skrabulakova; Naduvath Sudev

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The Thue colouring of a graph is a colouring such that the sequence of vertex colours of any path of even and finite length in $G$ is non-repetitive. The change in the Thue number, $\pi(G)$, as edges are iteratively removed from a graph $G$ is studied. The notion of the $\tau$-index denoted, $\tau(G)$, of a graph $G$ is introduced as well. $\tau(G)$ serves as a measure for the efficiency of edge deletion to reduce the Thue chromatic number of a graph.

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1601.02914

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Jun 27, 2018
06/18

by
Johan Kok; Naduvath Sudev; Chithra Sudev

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In this paper, we introduce the concept of curling subsequence of simple, finite and connected graphs. A curling subsequence is a maximal subsequence $C$ of the degree sequence of a simple connected graph $G$ for which the curling number $cn(G)$ corresponds to the curling number of the degree sequence per se and hence we call it the curling number of the graph $G$. A maximal degree subsequence with equal entries is called an identity subsequence. The number of identity curling subsequences in a...

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1506.00813

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4.0

Jun 29, 2018
06/18

by
N. K. Sudev; K. P. Chithra; Johan Kok

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Let $\mathcal{C} = \{c_1,c_2, c_3, \ldots,c_k\}$ be a certain type of proper $k$-colouring of a given graph $G$ and $\theta(c_i)$ denote the number of times a particular colour $c_i$ is assigned to the vertices of $G$. Then, the colouring sum of a given graph $G$ with respect to the colouring $\cC$, denoted by $\omega_{\cC}(G)$, is defined to be $\omega(\cC) = \sum\limits_{i=1}^{k}i\,\theta(c_i)$. The colouring sums such as $\chi$-chromatic sum, $\chi^+$-chromatic sum, $b$-chromatic sum,...

Topics: General Mathematics, Mathematics

Source: http://arxiv.org/abs/1606.00268

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9.0

Jun 28, 2018
06/18

by
Johan Kok; N. K. Sudev; K. P. Chithra

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Let $G(V, E)$ be a simple connected graph, with $|E| = \epsilon.$ In this paper, we define an edge-set graph $\mathcal G_G$ constructed from the graph $G$ such that any vertex $v_{s,i}$ of $\mathcal G_G$ corresponds to the $i$-th $s$-element subset of $E(G)$ and any two vertices $v_{s,i}, v_{k,m}$ of $\mathcal G_G$ are adjacent if and only if there is at least one edge in the edge-subset corresponding to $v_{s,i}$ which is adjacent to at least one edge in the edge-subset corresponding to...

Topics: Mathematics, General Mathematics

Source: http://arxiv.org/abs/1507.02546

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Jun 27, 2018
06/18

by
Johan Kok; N. K. Sudev; K. P. Chithra

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A \emph{primitive hole} of a graph $G$ is a cycle of length 3 in $G$. The number of primitive holes in a given graph $G$ is called the primitive hole number of the graph $G$. The primitive degree of a vertex $v$ of a given graph $G$ is the number of primitive holes incident on the vertex $v$. In this paper, we introduce the notion of Pythagorean holes of graphs and initiate some interesting results on Pythagorean holes in general as well as results in respect of set-graphs and Jaco graphs.

Topics: General Mathematics, Mathematics

Source: http://arxiv.org/abs/1504.07714

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22

Jun 26, 2018
06/18

by
Johan Kok; Susanth C; Sunny Joseph Kalayathankal

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Let $G^\rightarrow$ be a simple connected directed graph on $n \geq 2$ vertices and let $V^*$ be a non-empty subset of $V(G^\rightarrow)$ and denote the undirected subgraph induced by $V^*$ by, $\langle V^* \rangle.$ We show that the \emph{competition graph} of the Jaco graph $J_n(1), n \in \Bbb N, n \geq 5,$ denoted by $C(J_n(1))$ is given by:\\ \\ $C(J_n(1)) = \langle V^* \rangle_{V^* = \{v_i|3 \leq i \leq n-1\}} - \{v_iv_{m_i}| m_i = i + d^+_{J_n(1)}(v_i), 3 \leq i \leq n-2\} \cup \{v_1,...

Topics: Mathematics, Combinatorics

Source: http://arxiv.org/abs/1502.01824

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Jun 28, 2018
06/18

by
Johan Kok; Susanth C.; Sunny Joseph Kalayathankal

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We introduce the concept of a family of finite directed graphs (\emph{positive integer order,} $f(x) = mx + c; x,m \in \Bbb N$ and $c \in \Bbb N_0)$ which are directed graphs derived from an infinite directed graph called the $f(x)$-root digraph. The $f(x)$-root digraph has four fundamental properties which are; $V(J_\infty(f(x))) = \{v_i: i \in \Bbb N\}$ and, if $v_j$ is the head of an arc then the tail is always a vertex $v_i, i < j$ and, if $v_k$ for smallest $k \in \Bbb N$ is a tail...

Topics: Combinatorics, Mathematics

Source: http://arxiv.org/abs/1506.06538

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13

Jun 26, 2018
06/18

by
Johan Kok; Susanth C; Sunny Joseph Kalayathankal

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The concept of the brush number $b_r(G)$ was introduced for a simple connected undirected graph $G$. The concept will be applied to the Mycielskian graph $\mu(G)$ of a simple connected graph $G$ to find $b_r(\mu(G))$ in terms of an \emph{optimal orientation} of $G$. We prove a surprisingly simple general result for simple connected graphs on $n \geq 2$ vertices namely: $b_r(\mu(G))= b_r(\mu^{\rightarrow}(G)) = 2\sum\limits_{i=1}^{n}d^+_{G^{\rightarrow}_{b_r(G)}}(v_i).$

Topics: Mathematics, Combinatorics

Source: http://arxiv.org/abs/1501.03623

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Jun 26, 2018
06/18

by
Johan Kok; Susanth C; Sunny Joseph Kalayathankal

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The concept of the \emph{Gutman index}, denoted $Gut(G)$ was introduced for a connected undirected graph $G$. In this note we apply the concept to the underlying graphs of the family of Jaco graphs, (\emph{directed graphs by definition}), and describe a recursive formula for the \emph{Gutman index} $Gut(J^*_{n+1}(x)).$ We also determine the \emph{Gutman index} for the trivial \emph{edge-joint} between Jaco graphs.

Topics: Mathematics, Combinatorics

Source: http://arxiv.org/abs/1502.07093

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Jun 26, 2018
06/18

by
Johan Kok; Susanth C.; Sunny Joseph Kalayathankal

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The concept of the brush number $b_r(G)$ was introduced for a simple connected undirected graph $G$. The concept will be applied to the Mycielski Jaco graph $\mu(J_n(1)), n \in \Bbb N,$ in respect of an \emph{optimal orientation} of $J_n(1)$ associated with $b_r(J_n(1)).$ Further for the aforesaid, the concept of a \emph{brush centre} of a simple connected graph will be introduced. Because brushes themselves may be technology of kind, the technology in real world application will normally be...

Topics: Mathematics, Combinatorics

Source: http://arxiv.org/abs/1501.01381

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Jun 27, 2018
06/18

by
Johan Kok; K. P. Chithra; N. K. Sudev; C. Susanth

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A \textit{primitive hole} of a graph $G$ is a cycle of length $3$ in $G$. The number of primitive holes in a given graph $G$ is called the primitive hole number of that graph $G$. The primitive degree of a vertex $v$ of a given graph $G$ is the number of primitive holes incident on the vertex $v$. In this paper, we introduce the notion of set-graphs and study the properties and characteristics of set-graphs. We also check the primitive hole number and primitive degree of set-graphs. Interesting...

Topics: General Mathematics, Mathematics

Source: http://arxiv.org/abs/1504.02703

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Jun 29, 2018
06/18

by
Johan Kok; N. K. Sudev; K. P. Chithra; U. Mary

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In this paper, we introduce the notion of an energy graph as a simple, directed and vertex labeled graph $G$ such that the arcs $(v_i, v_j) \notin A(G)$ if $i > j$ for all distinct pairs $v_i,v_j$ and at least one vertex $v_k$ exists such that $d^-(v_k)=0$. Initially, equal amount of potential energy is allocated to certain vertices. Then, at a point of time these vertices transform the potential energy into kinetic energy and initiate transmission to head vertices. Upon reaching a head...

Topics: General Mathematics, Mathematics

Source: http://arxiv.org/abs/1607.00472

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4.0

Jun 29, 2018
06/18

by
Robert Plato

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This paper deals with Lavrentiev regularization for solving linear ill-posed problems, mostly with respect to accretive operators on Hilbert spaces. We present converse and saturation results which are an important part in regularization theory. As a byproduct we obtain a new result on the quasi-optimality of a posteriori parameter choices. Results in this paper are formulated in Banach spaces whenever possible.

Topics: Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1607.04879

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Jun 29, 2018
06/18

by
Robert Plato

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We study quadrature methods for solving Volterra integral equations of the first kind with smooth kernels under the presence of noise in the right-hand sides, with the quadrature methods being generated by linear multistep methods. The regularizing properties of an a priori choice of the step size are analyzed, with the smoothness of the involved functions carefully taken into consideration. The balancing principle as an adaptive choice of the step size is also studied. It is considered in a...

Topics: Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1604.08703

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Jun 30, 2018
06/18

by
Johan Kok; Paul Fisher; Bettina Wilkens; Mokhwetha Mabula; Vivian Mukungunugwa

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We introduce the concept of a family of finite directed graphs (order 1) which are directed graphs derived from an infinite directed graph (order 1), called the 1-root digraph. The 1-root digraph has four fundamental properties which are; $V(J_\infty(1)) = \{v_i|i \in \Bbb N\}$ and, if $v_j$ is the head of an edge (arc) then the tail is always a vertex $v_i, i n.$ Hence, trivially we have $d(v_i) \leq i$ for $i \in \Bbb N$. We present an interesting Fibonaccian-Zeckendorf result and present the...

Topics: Mathematics, Combinatorics

Source: http://arxiv.org/abs/1404.0484

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5.0

Jun 30, 2018
06/18

by
Johan Kok; Paul Fisher; Bettina Wilkens; Mokhwetha Mabula; Vivian Mukungunugwa

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We introduce the concept of a family of finite directed graphs (order a) which are directed graphs derived from an infinite directed graph (order a), called the a-root digraph. The a-root digraph has four fundamental properties which are; $V(J_\infty(a)) = \{v_i|i \in \Bbb N\}$ and, if $v_j$ is the head of an edge (arc) then the tail is always a vertex $v_i, i n.$ Hence, trivially we have $d(v_i) \leq ai$ for $i \in \Bbb N.$ We present an interesting Lucassian-Zeckendorf result and other...

Topics: Mathematics, Combinatorics

Source: http://arxiv.org/abs/1404.1714

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7.0

Jun 29, 2018
06/18

by
Johan Kok; N. K. Sudev; K. P. Chithra; K. A. Germina; U. Mary

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The concepts of linear Jaco graphs and Jaco-type graphs have been introduced as certain types of directed graphs with specifically defined adjacency conditions. The distinct difference between a pure Jaco graph and a Jaco-type graph is that for a pure Jaco graph, the total vertex degree $d(v)$ is well-defined, while for a Jaco-type graph the vertex out-degree $d^+(v)$ is well-defined. Hence, in the case of pure Jaco graphs a challenge is to determine $d^-(v)$ and $d^+(v)$ respectively and for...

Topics: General Mathematics, Mathematics

Source: http://arxiv.org/abs/1608.00856

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6.0

Jun 28, 2018
06/18

by
Susanth C; Sunny Joseph Kalayathankal; N. K. Sudev; K. P. Chithra; Johan Kok

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Given a finite nonempty sequence $S$ of integers, write it as $XY^k$, consisting of a prefix $X$ (which may possibly be empty), followed by $k$ copies of a non-empty string $Y$. Then, the greatest such integer $k$ is called the curling number of $S$ and is denoted by $cn(S)$. The concept of curling number of sequences has already been extended to the degree sequences of graphs to define the curling number of a graph. In this paper we study the curling number of graph powers, graph products and...

Topics: General Mathematics, Mathematics

Source: http://arxiv.org/abs/1512.01096

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7.0

Jun 28, 2018
06/18

by
N. K. Sudev; C. Susanth; K. P. Chithra; Johan Kok; Sunny Joseph Kalayathankal

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Let $S=S_1S_2S_3\ldots S_n$ be a finite string. Write $S$ in the form $XYY\ldots Y=XY^k$, consisting of a prefix $X$ (which may be empty), followed by $k$ copies of a non-empty string $Y$. Then, the greatest value of this integer $k$ is called the curling number of $S$ and is denoted by $cn(S)$. Let the degree sequence of the graph $G$ be written as a string of identity curling subsequences say, $X^{k_1}_1\circ X^{k_2}_2\circ X^{k_3}_3 \ldots \circ X^{k_l}_l$. The compound curling number of...

Topics: General Mathematics, Mathematics

Source: http://arxiv.org/abs/1510.01271

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16

Jun 28, 2018
06/18

by
Susanth C.; Sunny Joseph Kalayathankal; N. K. Sudev; K. P. Chithra; Johan Kok

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Given a finite nonempty sequence $S$ of integers, write it as $XY^k$, where $Y^k$ is a power of greatest exponent that is a suffix of $S$: this $k$ is the curling number of $S$. The concept of curling number of sequences has already been extended to the degree sequences of graphs to define the curling number of a graph. In this paper we study the curling number of graph powers, graph products and certain other graph operations.

Topics: Mathematics, General Mathematics

Source: http://arxiv.org/abs/1509.00220

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53

Jul 26, 2018
07/18

by
Plato; Kyriakos Zambas

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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος Ε

Topics: JC, State, The -- Early works to 1800, Political science -- Early works to 1800

Source: https://www.gutenberg.org/ebooks/36262

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35

Jul 26, 2018
07/18

by
Plato; Kyriakos Zambas

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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος Δ

Topics: State, The -- Early works to 1800, JC, Political science -- Early works to 1800

Source: https://www.gutenberg.org/ebooks/36284

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48

Jul 26, 2018
07/18

by
Plato; Kyriakos Zambas

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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος Γ

Topics: JC, Political science -- Early works to 1800, State, The -- Early works to 1800

Source: https://www.gutenberg.org/ebooks/36191

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53

Jul 26, 2018
07/18

by
Plato; Kyriakos Zambas

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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος Α

Topics: Political science -- Early works to 1800, JC, State, The -- Early works to 1800

Source: https://www.gutenberg.org/ebooks/36088

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36

Jul 26, 2018
07/18

by
Kyriakos Zambas; Plato

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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος B

Topics: Political science -- Early works to 1800, JC, State, The -- Early works to 1800

Source: https://www.gutenberg.org/ebooks/36116