A fuzzy graph is one of the versatile application tools in the field of mathematics, which allows the user to easily describe the fuzzy relation between any objects. The nature of fuzziness is favorable for any environment, which supports to predict the problem and solving it. Fuzzy graphs are beneficial to give more precision and flexibility to the system as compared to the classical model (i.e.,) crisp theory. A topological index is a numerical quantity for the structural graph of the molecule and it can be represented through Graph theory. Moreover, its application not only in the field of chemistry can also be applied in areas including computer science, networking, etc. A lot of topological indices are available in chemical-graph theory and H. Wiener proposed the first index to estimate the boiling point of alkanes called ‘Wiener index’. Many topological indices exist only in the crisp but it’s new to the fuzzy graph environment. The main aim of this paper is to define the topological indices in fuzzy graphs. Here, indices defined in fuzzy graphs are Modified Wiener index, Hyper Wiener index, Schultz index, Gutman index, Zagreb indices, Harmonic index, and Randić index with illustrations. Bounds for some of the indices are proved. The algorithms for distance matrix and MWI are shown. Finally, the application of these indices is discussed.

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Tapioca starch is one among the staple foods for human, especially in Tamil Nadu, India. The sago pearls can be produced from tapioca starch through a preparation process (PP). During PP failure may occur in the processing unit, and it may affect the quality of sago pearls, which is a major issue in sago industries. Hence, in our study, a risk assessment method based on fuzzy petri net (FPN) is proposed for the assessment of sago PP. FPN is a graphical mathematical modeling, which provides the basis for modeling the systems with uncertain and inadequate information. Evaluation of weights for places and transitions is a major concern in FPN models, to overcome this limitation, this study proposes a fuzzy analytic hierarchy process to evaluate the weights of the places and transitions in FPN. A FPN model for sago processing unit is defined and by using a matrix‐based reasoning algorithm, risk values for various levels and for the entire system is evaluated. Our method is the one of the reliable methods in risk evaluation of the sago mills and production factories. Practical Applications A specific case study on sago manufacturing process in a factory at Namakkal district, Tamil Nadu is adopted to project the significance of the proposed FPN model with FAHP. For evaluation process, risk factors and token value of the risk factors for FPN model are acquired from the experts in the industry. From the evaluated results of the study attained from the proposed models, it is observed that the results are more consistent with the actual condition. Although the results obtained from the proposed are too reliable with the actual condition of the factory, the proposed FPN‐based risk evaluation methodology for the sago manufacturing process offers reasonable support to the Fault diagnosing unit to avoid the failure in the industry.

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