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Formulating and Solving Integer Programs
Published in Craig A. Tovey, Linear Optimization and Duality, 2020
The graph automorphism problem is to determine whether there exists an isomorphism of a graph G=(V,E) to itself, other than the trivial identity mapping σ(v)=v∀v∈V. Model this problem as an IP. (Caution: The automorphism may map some vertices to themselves, as long as it does not map all vertices to themselves.)
Structure and Representation
Published in Jonathan L. Gross, Jay Yellen, Mark Anderson, Graph Theory and Its Applications, 2018
Jonathan L. Gross, Jay Yellen, Mark Anderson
Example 2.2.2: The graph K1,3 has six automorphisms. Each of them is realizable by a rotation or reflection of the drawing in Figure 2.2.1. For instance, a 120° clockwise rotation of the figure corresponds to the graph automorphism with vertex-permutation (x)(u v w) and edge-permutation (a b c). Also, reflection through the vertical axis corresponds to the graph automorphism with vertex-permutation (x)(u)(v w) and edge-permutation (a)(b c). The following table lists all the automorphisms of K1,3 and their corresponding vertex- and edge-permutations.
Two-dimensional coronene fractal structures: topological entropy measures, energetics, NMR and ESR spectroscopic patterns and existence of isentropic structures
Published in Molecular Physics, 2022
Micheal Arockiaraj, Joseph Jency, Jessie Abraham, S. Ruth Julie Kavitha, Krishnan Balasubramanian
For example, there are 2424 DDSV tuples of variable length for the two isentropic pairs shown in Figure 6. Note that the DDSV for each vertex is a vector of variable length, as the end point of the sequence for each vertex is determined by its eccentricity. Hence, two equivalent vertices under the graph automorphism carry the same DDSV, and hence we employ DDSV as a tool for computing the equivalence classes of vertices based on their DDSV. We employed a concatenation algorithm that joins the components of the p-tuple to generate a very long integral identification code for each vertex. We then compare the concatenated vertex identification code to generate the vertex partitioning from the DDSV vector.
Topological study on degree based molecular descriptors of fullerene cages
Published in Molecular Physics, 2023
Tony Augustine, S. Roy, J. Sahaya Vijay, Jain Maria Thomas, P. Shanmugam
A number, a polynomial, a series of integers, or a matrix representing the whole graph may be used to identify a graph. Each of these representations aims to be specified explicitly for that graph. A topological index is a numerical value that describes the topology of a graph and is unaffected by graph automorphism. Topological indices may be divided into many broad categories, including degree-based topological indices, distance-based topological indices, polynomials linked to counting [52]. Degree-based topological indices are of tremendous significance and are especially important in chemical graph theory and chemistry.
The Garden of Eden theorem for cellular automata on group sets
Published in International Journal of Parallel, Emergent and Distributed Systems, 2019
The group G encodes some graph automorphisms of M, more precisely, the component encodes the translational automorphisms and the component R the rotational ones that stabilise the origin. For example, the element encodes the anticlockwise rotation about the origin through 90°, followed by a translation by , which is the anticlockwise rotation about a through 90°; see Figure 2 for further examples. In general, for each vertex and each angle , the anticlockwise rotation about m through r°, is the graph automorphism , which is encoded by . The map embeds the group G into the graph-automorphism group of M. Let be the neutral element of and, for each vertex , let be the translation (m, 0). The tuple is a coordinate system for . The stabiliser of under is the set , the quotient group is isomorphic to the group by , and, under this isomorphism, the right quotient group semi-action of on M is the map , .