Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 11 3 2026 09 01 The monophonic pebbling number of neural networks with generalized algorithm and their applications 827 842 14848 10.22049/cco.2024.29481.2017 EN K.C. Kavitha Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology Vellore - 632014, Tamil Nadu, India S. Jagatheswari Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology Vellore - 632014, Tamil Nadu, India Journal Article 2024 02 24 Consider a graph $\sigma$(V, E) with nodes V and edges E is a connected graph with some pebbles scattered over its nodes V. By removal of two pebbles from one node and placing one pebble to an adjacent node is a pebbling move. A monophonic pebbling number, $\lambda_{M}(\sigma, v)$, of a node v of a graph $\sigma$ is the least number $m$ such that minimum of one pebble could be shifted to v by a sequence of pebbling shifts for any distribution of $\lambda_{M}(\sigma, v)$ pebbles on the nodes of $\sigma$ using monophonic path. A link between the nodes x and y is an x-y path which consists of no chords and is monophonic. The monophonic pebbling number of a graph $\sigma$ is the highest $\lambda_{M}(\sigma, v)$ among all the nodes notated as $\lambda_{M}(\sigma)$. For the first time, we calculate the monophonic pebbling number on families of neural networks such as probabilistic neural networks(PNNs),  convolutional neural networks(CVNNs), modular neural networks(MNNs), generalized regression neural networks(GRNNs) and Hopfield neural networks(HNNs) and discuss their applications. We give the generalized algorithm to find the monophonic pebbling number of any graph $\sigma$. https://comb-opt.azaruniv.ac.ir/article_14848_4121efbad41dff3eae036d4be2811bc4.pdf