Isomorphism on generalized fuzzy graphs and image visualizations
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Isomorphism on generalized fuzzy graphs and image visualizations Sovan Samanta1
· Biswajit Sarkar2
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The graph theory is being used for representation in networks and chemical atomic structures very frequently. However, these days, uncertainties are imposed on such networks. Isomorphism in generalized fuzzy graphs has been introduced here to capture the similarity of uncertainties in different networks. Homomorphism, weak isomorphism, co-weak isomorphism and nearly isomorphism are defined with examples. Also, an application of image visualization is described. Keywords Fuzzy graphs · Generalized fuzzy graphs · Isomorphism · Network · Image visualization
1 Introduction The ambiguity or haziness of the parameters in the social networks is present in all the networks properly. A social network, a perfect example, may be designed as a graph where nodes/vertices denote an account (individual, institution, etc.), and edges denote the relationship between the nodes. If the relationships among nodes are measured as genuine according to the recurrence of interactions among the nodes, fuzziness can be combined for such design. This problems and similar other problems commence to define fuzzy graphs. Kauffman (1973) gave the original definition of a fuzzy graph in 1973. However, in 1975, it was Rosenfeld (1975) who recognized fuzzy relations on fuzzy sets and advanced the theory of fuzzy graphs. Using this idea of a fuzzy graph, Koczy (1992) adopted fuzzy graphs in the calculation and optimization of networks. For any further details of fuzzy graphs, readers may examine Mathew and Sunitha (2009, 2013), McAllister (1988), Mordeson and Nair (2000), Pramanik et al. (2017) and Samanta and Pal (2013, 2015).
Communicated by A. Di Nola.
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Biswajit Sarkar [email protected] Sovan Samanta [email protected]
1
Tamralipta Mahavidyalaya, Tamluk, WB 721636, India
2
Department of Industrial Engineering, Yonsei University, 50 Yonsei-ro, Sinchon-dong, Seodaemun-gu, Seoul 03722, South Korea
There are numerous types of fuzzy graphs accessible in the literature. Intuitionistic fuzzy graph (Parvathi and Karunambigai 2006), interval-valued fuzzy graphs (Akram and Dudek 2011) and bipolar fuzzy graphs (Akram 2011) are few of them. In all the fuzzy graphs, there is a general property that edge membership value is smaller or equal to the minimum of its membership values of end vertices. Suppose a social network is to be designed as fuzzy graphs. Here, all units are taken as fuzzy nodes/vertices. The membership values of the vertices depend on several parameters like authenticity or duration of the accounts from creation. Suppose the membership values of nodes are estimated according to the sources of knowledge. Fuzzy edges represent the relationship between the units. The membership value is calculated according to the transfer of knowledge. However, transfer of knowledge may be greater than one of the social actors/units as a more knowledgeable per
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