• Earthquakes and Structures
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• 제10권1호
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• pp.25-51
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• 2016

In this study, graph product rules are applied to the dynamic analysis of regular skeletal structures. Graph product rules have recently been utilized in structural mechanics as a powerful tool for eigensolution of symmetric and regular skeletal structures. A structure is called regular if its model is a graph product. In the first part of this paper, the formulation of time history dynamic analysis of regular structures under seismic excitation is derived using graph product rules. This formulation can generally be utilized for efficient linear elastic dynamic analysis using vibration modes. The second part comprises of random vibration analysis of regular skeletal structures via canonical forms and closed-form eigensolution of matrices containing special patterns for symmetric structures. In this part, the formulations are developed for dynamic analysis of structures subjected to random seismic excitation in frequency domain. In all the proposed methods, eigensolution of the problems is achieved with less computational effort due to incorporating graph product rules and canonical forms for symmetric and cyclically symmetric structures.

• Journal of applied mathematics & informatics
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• 제37권3_4호
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• pp.163-176
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• 2019

Given a distribution of pebbles on the vertices of a connected graph G, a pebbling move is defined as the removal of two pebbles from some vertex and the placement of one of those pebbles at an adjacent vertex. The t-pebbling number, $f_t(G)$, of a connected graph G, is the smallest positive integer such that from every placement of $f_t(G)$ pebbles, t pebbles can be moved to any specified vertex by a sequence of pebbling moves. A graph G has the 2t-pebbling property if for any distribution with more than $2f_t(G)$ - q pebbles, where q is the number of vertices with at least one pebble, it is possible, using the sequence of pebbling moves, to put 2t pebbles on any vertex. In this paper, we determine the t-pebbling number for the middle graph of a complete binary tree $M(B_h)$ and we show that the middle graph of a complete binary tree $M(B_h)$ satisfies the 2t-pebbling property.

• Journal of Applied and Pure Mathematics
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• 제4권3_4호
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• pp.85-97
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• 2022

Let a graph G = (V, E) be a (p, q) graph. Define $${\rho}\;=\;\{\array{{\frac{p}{2}}&p\text{ is even}\\{\frac{p-1}{2}}\;&p\text{ is odd,}}$$ and M = {±1, ±2, ⋯ ± 𝜌} called the set of labels. Consider a mapping λ : V → M by assigning different labels in M to the different elements of V when p is even and different labels in M to p - 1 elements of V and repeating a label for the remaining one vertex when p is odd. The labeling as defined above is said to be a pair mean cordial labeling if for each edge uv of G, there exists a labeling $\frac{{\lambda}(u)+{\lambda}(v)}{2}$ if λ(u) + λ(v) is even and $\frac{{\lambda}(u)+{\lambda}(v)+1}{2}$ if λ(u) + λ(v) is odd such that ${\mid}\bar{\mathbb{S}}_{{\lambda}_1}-\bar{\mathbb{S}}_{{\lambda}^c_1}{\mid}{\leq}1$ where $\bar{\mathbb{S}}_{{\lambda}_1}$ and $\bar{\mathbb{S}}_{{\lambda}^c_1}$ respectively denote the number of edges labeled with 1 and the number of edges not labeled with 1. A graph G for which there exists a pair mean cordial labeling is called a pair mean cordial graph. In this paper, we investigate the pair mean cordial labeling of graphs which are obtained from path and cycle.

• Journal of applied mathematics & informatics
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• 제30권3_4호
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• pp.353-364
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• 2012

The Wiener polarity index$W_p(G)$ of a graph G of order $n$ is the number of unordered pairs of vertices $u$ and $v$ of G such that the distance $d_G(u,v)$ between $u$ and $v$ is 3. In this paper the Wiener polarity index of some graph operations are computed. As an application of our results, the Wiener polarity index of a polybuckyball fullerene and $C_4$ nanotubes and nanotori are computed.

• Journal of Multimedia Information System
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• 제2권3호
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• pp.263-274
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• 2015

The subject of this work is the complex discrete systems simulation special features with the aid of dynamic graph models. The proposed simulation technique allows to determine the ways for tasks solutions in terms of discrete systems analysis and synthesis of various complication: one-dimensional and multidimensional, steady and unstable, with the pulse elements abnormal operating mode and others. Often complex control systems analysis and synthesis task solutions, via classical approach comes out to be insolvent, because of the computational problems. The application of graph models allows to perform clear and strict characterization and computer procedures automation. The optimal controls synthesis algorithm presented in this paper, transferring the discrete system from target initial state to target final state within the minimum time, allows to consider the zero initial conditions systems, with the initial potential energy, with the control actions limitations and complex pulse elements operating mode.

• Journal of applied mathematics & informatics
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• 제26권5_6호
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• pp.1085-1100
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• 2008

A set D of vertices of a graph G = (V(G),E(G)) is called a dominating set if every vertex of V(G) - D is adjacent to at least one element of D. The domination number of G, denoted by ${\gamma}(G)$, is the size of its smallest dominating set. Haynes et al.[5] present a conjecture: For any graph G with ${\delta}(G){\geq}k$,$\gamma(G){\leq}\frac{k}{3k-1}n$. When $k\;{\neq}\;6$, the conjecture was proved in [7], [8], [10], [12] and [13] respectively. In this paper we prove that every graph G on n vertices with ${\delta}(G)\;{\geq}\;6$ has a dominating set of order at most $\frac{6}{17}n$. Thus the conjecture was completely proved.

• Journal of applied mathematics & informatics
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• 제30권5_6호
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• pp.913-923
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• 2012

The Odd-Even graceful labeling of a graph G with $q$ edges means that there is an injection $f:V (G)$ to $\{1,3,5,{\cdots},2q+1\}$ such that, when each edge $uv$ is assigned the label ${\mid}f(u)-f(v){\mid}$, the resulting edge labels are $\{2,4,6,{\cdots},2q\}$. A graph which admits an odd-even graceful labeling is called an odd-even graceful graph. In this paper, we prove that some well known graphs namely $P_n$, $P_n^+$, $K_{1,n}$, $K_{1,2,n}$, $K_{m,n}$, $B_{m,n}$ are Odd-Even graceful.

• Journal of applied mathematics & informatics
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• 제35권1_2호
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• pp.25-32
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• 2017

Let G be a graph and k is a positive integer. Hammack and Livesay in [The inner power of a graph, Ars Math. Contemp., 3 (2010), no. 2, 193-199] introduced a new graph operation $G^{(k)}$, called the $k^{th}$ inner power of G. In this paper, it is proved that if G is bipartite then $G^{(2)}$ has exactly three components such that one of them is bipartite and two others are isomorphic. As a consequence the edge frustration index of $G^{(2)}$ is computed based on the same values as for the original graph G. We also compute the first and second Zagreb indices and coindices of $G^{(2)}$.

• Journal of applied mathematics & informatics
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• 제39권3_4호
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• pp.497-506
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• 2021

Let G be a graph. Let f : V (G) → {0, 1, …, k - 1} be a function where k ∈ ℕ and k > 1. For each edge uv, assign the label $f(uv)={\lceil}{\frac{f(u)+f(v)}{2}}{\rceil}$. f is called k-total mean cordial labeling of G if ${\mid}t_{mf}(i)-t_{mf}(j){\mid}{\leq}1$, for all i, j ∈ {0, 1, …, k - 1}, where t_mf (x) denotes the total number of vertices and edges labelled with x, x ∈ {0, 1, …, k-1}. A graph with admit a k-total mean cordial labeling is called k-total mean cordial graph.

• 한국컴퓨터정보학회논문지
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• 제23권12호
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• pp.57-64
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• 2018

In this paper, a method to measure similarity between two graphs is proposed, which is based on centralities of the graphs. The similarity between two graphs $G_1$ and $G_2$ is defined by the difference of distance($G_1$, $G_{R_1}$) and distance($G_2$, $G_{R_2}$), where $G_{R_1}$ and $G_{R_2}$ are set of random graphs that have the same number of nodes and edges as $G_1$ and $G_2$, respectively. Each distance ($G_*$, $G_{R_*}$) is obtained by comparing centralities of $G_*$ and $G_{R_*}$. Through the computational experiments, we show that it is possible to compare graphs regardless of the number of vertices or edges of the graphs. Also, it is possible to identify and classify the properties of the graphs by measuring and comparing similarities between two graphs.