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Polynomials and graph homomorphisms
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
Delia Garijo, Andrew Goodall, Jaroslav Nešetřil, Guus Regts
A graph invariant is a function on graphs invariant under isomorphism. A graph invariant f defines a partition on graphs in which G and G′ are equivalent when f(G)=f(G′), in which case we write G~fG′. The partition of graphs by f is a coarsening of the partition of graphs by isomorphism. A graph invariant determines G up to isomorphism precisely when the equivalence class containing G under f is the isomorphism class of G, that is, G~fG′ implies G≅G′. The graph G is then said to be f-unique. The subject of Chapter 6 is Tutte uniqueness, that is, the question of which graphs are uniquely determined up to isomorphism by their Tutte polynomials.
A comparison of centrality measures and their role in controlling the spread in epidemic networks
Published in International Journal of Control, 2023
Ekaterina Dudkina, Michelangelo Bin, Jane Breen, Emanuele Crisostomi, Pietro Ferraro, Steve Kirkland, Jakub Mareček, Roderick Murray-Smith, Thomas Parisini, Lewi Stone, Serife Yilmaz, Robert Shorten
Many connectivity measures exist which provide a single, numerical value describing the ‘connectedness’ of a network in some way. Such measures can be easily extended to measure the criticality of a single node i in a graph by inferring the criticality of the node from the change in criticality of the network after the node is removed from the network. This is described as follows in Fouss et al. (2016): given some graph invariant measuring the connectedness of the graph G in some manner, define We include several such measures of node centrality here for consideration in later simulations.
Matching polynomial coefficients and the Hosoya indices of poly(p-phenylene) graphs
Published in Molecular Physics, 2018
Tapanendu Ghosh, Sukanya Mondal, Bholanath Mandal
The algorithms developed here for calculating the MPs and hence the Hosoya indices (Z) of linear and cylindrical PPP graphs are very convenient to use for this purpose. The MP of an acyclic graph is just equal to the CP [6,7,19,33–38], another important graph invariant obtained in expanding the secular determinant in Hückel molecular orbital formalism under some approximations [6,7,33]. Although for a cyclic graph the CP is not equal to the MP but can be obtained from it by incorporating the contribution of the cycle(s) in the graph. The MP can be imagined to be the arithmetic mean of the CPs of the corresponding Hückel and Möbius chemical graphs [39]. One of the important uses of MP is to calculate one of the important topological indices, the Hosoya index [1,6,7,30,31,40]. Beside this, it is found to be used in determining the number of Kekulé resonance structure, Dewar resonance structure of polycyclic aromatic compounds [6,7,33,41,42] and to obtain the topological resonance energy [2,3,6,7,39], a measure of the energetic chemical aromaticity in the Hückel level of theory. Moreover, the graphs considered here in this study are important polycyclic aromatic hydrocarbons with remarkable mechanical, electronic and optical properties [8–16,23–29].