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Universal Absolute Mean Graceful Graphs
Published in N. P. Shrimali, Nita H. Shah, Recent Advancements in Graph Theory, 2020
P. Chudasama Hiren, K. Jadeja Divya
Kaneria and Chudasama [3] applied more liberty to vertex labeling in graceful labeling and introduced absolute mean graceful labeling. Kaneria and Chudasama [3] proved that all path Pn, cycle Cn, complete bipartite graph Km,n, grid graph Pm × Pn, step grid graph Stn and double step grid graph DStn are absolute mean graceful graphs. Makadia et al. [4] introduced the concept of the graceful centre and universal graceful graph. The present chapter shows the way ahead to new researches for the concept of universal graceful graphs. For conceptual parts and notations, we refer to J. A. Gallian [1] and Harary [2].
The Directional Discrete Cosine Transform
Published in Humberto Ochoa-Domínguez, K. R. Rao, Discrete Cosine Transform, 2019
Humberto Ochoa-Domínguez, K. R. Rao
Then, the 2D eigenbasis2D eigenbasis are the 2D DCT basis images2D DCT basis images and can be obtained by the Kronecker productKronecker product of the 1D eigenbasis1D eigenbasis by using (7.13). The sum of their corresponding eigenvalues are useful to determine the multiplicity. In case that the resulting λk,l = λl,k their corresponding eigenvectors are still linearly independent, because the Kronecker product is not commutative. Therefore, the geometric multiplicityGeometric multiplicity is equal to the algebraic multiplicityAlgebraic multiplicity. In this sense, Fracastoro et al. [137] give the following proposition, The 2D-DCT is not the unique eigenbasis for the LaplacianEigenbasis for the Laplacian of a square grid graph of a square grid graph [137].
Physical Design Automation
Published in M. Michael Vai, Vlsi Design, 2017
Maze routing is a general-purpose routing algorithm which finds a path between two points in the routing space. The routing region is modeled as a grid graph. A routing surface is considered a collection of square cells arranged into a rectangular array. The size of the square cells is defined according to the design rules so that wires can be routed through adjacent cells without violating the rules for wire width and spacing. Cells that are already occupied by wiring, building blocks, etc. are marked as blocked. Fig. 10.10 shows the grid model of an example routing space. The blocked cells are shaded in the grid model.
Longest (s, t)-paths in L-shaped grid graphs
Published in Optimization Methods and Software, 2019
Fatemeh Keshavarz-Kohjerdi, Alireza Bagheri
A grid graph is a graph in which vertices lie on integer coordinates and edges connect vertices that are separated by a distance of one. A solid grid graph is a grid graph without holes. A rectangular grid graph is the subgraph of (the infinite grid graph) induced by , where and are x and y coordinates of v respectively. An L-shaped grid graph is a grid graph obtained from a rectangular grid graph by removing its subgraph from the upper right (or bottom left) corner.
A linear-time algorithm for finding Hamiltonian cycles in rectangular grid graphs with two rectangular holes
Published in Optimization Methods and Software, 2023
Fatemeh Keshavarz-Kohjerdi, Alireza Bagheri
A grid graph is an undirected graph that can be embedded in the plane such that its vertices lie on the integer coordinates and two vertices are connected by an edge if and only if the Euclidean distance between them is equal to 1 (see Figure 1(a)). A solid grid graph is a grid graph without holes (see Figure 1(b)). A rectangular grid graph is a grid graph bounded by an axis-parallel rectangle (see Figure 1(c)).