• The Pure and Applied Mathematics
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• v.27 no.4
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• pp.171-186
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• 2020

Long back in 1972, it was shown that the sum of the squares of vertex degrees and the sum of cubes of vertex degrees of a molecular graph both have large correlations with total 𝜋-electron energy of the molecule. Later on, the sum of squares of vertex degrees was named as first Zagreb index and became one of the most studied molecular graph parameter in the field of chemical graph theory. Whereas, the other sum remained almost unnoticed until recently except for a few occasions. Thus it got the name "forgotten" index or F-index. This paper investigates extremal graphs with respect to F-index among the class of bicyclic graphs with n vertices and k pendant vertices, 0 ≤ k ≤ n - 4. As consequences, we obtain the bicyclic graphs with largest and smallest F-indices.

• Journal of applied mathematics & informatics
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• v.31 no.5_6
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• pp.595-608
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• 2013

Let G = (V, E) be a simple connected graph. The sets of vertices and edges of G are denoted by V = V (G) and E = E(G), respectively. In such a simple molecular graph, vertices represent atoms and edges represent bonds. The distance between the vertices $u$ and $v$ in V (G) of graph G is the number of edges in a shortest path connecting them, we denote by $d(u,v)$. In graph theory, we have many invariant polynomials for a graph G. In this paper, we focus on the Schultz polynomial, Modified Schultz polynomial, Hosoya polynomial and their topological indices of a molecular graph circumcoronene series of benzenoid $H_k$ and specially third member from this family. $H_3$ is a basic member from the circumcoronene series of benzenoid and its conclusions are base calculations for the Schultz polynomial and Hosoya polynomial of the circumcoronene series of benzenoid $H_k$ ($k{\geq}3$).

• Proceedings of the IEEK Conference
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• 2003.11b
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• pp.11-14
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• 2003

In interconnection network for parallel processing, the cycle partitioning problem for parallel transmission with faulty vertieces or edges is very important. In this paper, we assume that k($\leq$m-1) edges do not share any vertices of m dimension hypercube Q$_{m}$ and show that it is possible to construct a cycle set which consists of k cycles covering all the vertices of the hypercube and one cycle including one of the given edges. This cycle set can be used to parallel transmission between two vertices joined by faulty edges.s.

• Journal of KIISE:Computer Systems and Theory
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• v.34 no.5_6
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• pp.176-186
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• 2007

An interconnection network can be represented as a graph where a vertex corresponds to a node and an edge corresponds to a link. The diameter of an interconnection network is the maximum length of the shortest paths between all pairs of vertices. The fault diameter of an interconnection network G is the maximum length of the shortest paths between all two fault-free vertices when there are $_k(G)-1$ or less faulty vertices, where $_k(G)$ is the connectivity of G. The fault diameter of an R-regular graph G with diameter of 3 or more and connectivity ${\tau}$ is at least diam(G)+1 where diam(G) is the diameter of G. We show that the fault diameter of a 2-dimensional $m{\times}n$ torus with $m,n{\geq}3$ is max(m,n) if m=3 or n=3; otherwise, the fault diameter is equal to its diameter plus 1. We also show that in $d({\geq}3)$-dimensional $k_1{\times}k_2{\times}{\cdots}{\times}k_d$ torus with each $k_i{\geq}3$, there are 2d mutually disjoint paths joining any two vertices such that the lengths of all these paths are at most diameter+1. The paths joining two vertices u and v are called to be mutually disjoint if the common vertices on these paths are u and v. Using these mutually disjoint paths, we show that the fault diameter of $d({\geq}3)$-dimensional $k_1{\times}k_2{\times}{\cdots}{\times}k_d$ totus with each $k_i{\geq}3$ is equal to its diameter plus 1.

• Communications of the Korean Mathematical Society
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• v.35 no.1
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• pp.1-12
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• 2020

A pathos block line cut-vertex graph of a tree T, written P BL_c(T), is a graph whose vertices are the blocks, cut-vertices, and paths of a pathos of T, with two vertices of P BL_c(T) adjacent whenever the corresponding blocks of T have a vertex in common or the edge lies on the corresponding path of the pathos or one corresponds to a block B_i of T and the other corresponds to a cut-vertex c_j of T such that c_j is in B_i; two distinct pathos vertices P_m and P_n of P BL_c(T) are adjacent whenever the corresponding paths of the pathos P_m(v_i, v_j) and P_n(v_k, v_l) have a common vertex. We study the properties of P BL_c(T) and present the characterization of graphs whose P BL_c(T) are planar; outerplanar; maximal outerplanar; minimally nonouterplanar; eulerian; and hamiltonian. We further show that for any tree T, the crossing number of P BL_c(T) can never be one.

• Journal of the Korean Mathematical Society
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• v.49 no.1
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• pp.85-98
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• 2012

Let R be a commutative ring and I its proper ideal, let S(I) be the set of all elements of R that are not prime to I. Here we introduce and study the total graph of a commutative ring R with respect to proper ideal I, denoted by T(${\Gamma}_I(R)$). It is the (undirected) graph with all elements of R as vertices, and for distinct x, y ${\in}$ R, the vertices x and y are adjacent if and only if x + y ${\in}$ S(I). The total graph of a commutative ring, that denoted by T(${\Gamma}(R)$), is the graph where the vertices are all elements of R and where there is an undirected edge between two distinct vertices x and y if and only if x + y ${\in}$ Z(R) which is due to Anderson and Badawi [2]. In the case I = {0}, $T({\Gamma}_I(R))=T({\Gamma}(R))$; this is an important result on the definition.

• Bulletin of the Korean Mathematical Society
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• v.44 no.2
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• pp.259-263
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• 2007

Which spheres are shellable?[2]. We present one of them which is the k-tree with n-labeled vertices. We found that the number of ways to shell the k-dimensional trees on n-labeled vertices is $$\frac{n!}{(k+1)!}(nk-k^2-k+1)!k$$.

• Journal of the Korean Mathematical Society
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• v.51 no.1
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• pp.189-202
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• 2014

An element r of a commutative semiring R with identity is said to be identity-summand if there exists $1{\neq}a{\in}R$ such that r+a = 1. In this paper, we introduce and investigate the identity-summand graph of R, denoted by ${\Gamma}(R)$. It is the (undirected) graph whose vertices are the non-identity identity-summands of R with two distinct vertices joint by an edge when the sum of the vertices is 1. The basic properties and possible structures of the graph ${\Gamma}(R)$ are studied.

• Journal of the Korea Society of Computer and Information
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• v.22 no.3
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• pp.61-67
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• 2017

This paper analyzes the robustness of the network based on the centrality of vertices in the graph. In this paper, a random graph is generated and a modified graph is constructed by adding or removing vertices or edges in the generated random graph. And then we analyze the robustness of the graph by observing changes in the centrality of the random graph and the modified graph. In the process modifying a graph, we changes some parts of the graph, which has high values of centralities, not in the whole. We study how these additional changes affect the robustness of the graph when changes occurring a group that has higher centralities than in the whole.

• Proceedings of the IEEK Conference
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• 2006.06a
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• pp.745-746
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• 2006

In this paper, we propose a new predictive coding scheme for color data of three-dimensional (3-D) mesh models. We exploit connectivity and geometry information to improve coding efficiency. After ordering all vertices in a 3-D mesh model with a vertex traversal technique, we employ a geometry predictor to compress the color data. The predicted color can be acquired by a weighted sum of reconstructed colors for adjacent vertices using both angles and distances between the current vertex and adjacent vertices.