Vulnerability parameters of fuzzy graphs: Fuzzy integrity and fuzzy scatteringnumber

Vulnerability is a measure of the durability of the network in the face of damage that could lead to a reduction or complete loss of a certain functionality of the network. Parameters were studied to investigate the vulnerabilities of networks modeled more closely with fuzzy graphs. In this thesis, fuzzy integrity I_f fuzzy scattering number 〖sc〗_f which are among the vulnerability parameters are defined in accordance with the structure of fuzzy graphs. The basic concepts of these parameters, maximum strength of connectedness m_f (G-S) and decreasing strength of connectedness ω_f (G-S) are expressed based on the strength of connectedness between the nodes of the graph. It is explained in detail on examples and with their applicability in daily life. General results are given by applying the node connectivity defined on the basis of the decrease in strength of connectedness in fuzzy graphs to customized fuzzy cycle graphs. Fuzzy integrity and fuzzy scattering number parameters are applied to fuzzy cycle graph with a single α-strong arc, locamin cycle, self-centered fuzzy cycle, regular fuzzy cycle and a fuzzy graph with a single weakest arc whose underlying graph is a cycle and general results are obtained. Besides, a general result is found for the fuzzy scattering number of fuzzy cycle graphs. Also different fuzzy wheels are constructed, labeled with the vertices u_1,u_2,…,u_(n-1) on the fuzzy cycle C_(n-1) and a central vertex u_n, by choosing all the membership values μ(u_n,u_i ) equal for i=1,…,n-1 and by binding relationship between μ(u_n,u_i ) and μ(e) specific criteria to the studied parameters are applied to these wheels and fuzzy star graphs and general results are obtained.