Study of mean-first-passage time and Kemeny's constant of a random walk by normalized Laplacian matrices of a penta-chain network

The mean first-passage time (MFPT), which refers to the expected time it takes for a system to reach a state j given its current state i, that is t(ji), falls under the fundamental theory of Markov processes. The set of mean first-passage time (MFPT) among the positions of a Markov process expands fundamental assumptions of the system's kinetics through their relation to the spectrum and eigenvectors of the transition matrix, and the moderation times of the random walker which all are of specific computational position. The explicit and precise computation of MFPT of random walks on networks can typically be highly challenging for networks with more than a few nodes, since they translate the global properties of the random walkers and the network they explore. On the other hand, in a connected network, the Kemeny's constant (KC) gives the expected time of a random walk from an arbitrary vertex x to reach a randomly chosen vertex y. The KC is interpreted as a measure of the connectivity level of a network, indicating how effectively the network is interconnected. The KC is an inspiring and helpful quantifier due to its rich applications, mostly in Markov's chain. In the literature, there are multiple approaches to represent the complete matrix of MFPT. Among them, one widely used and traditional method is to employ the normalized Laplacian matrix. This study presents a new graph spectrum-based approach to compute the MFPT and KC of random walks on penta-chain network ('O). By using the decomposition theorem of normalized Laplacian polynomial, we computed the normalized Laplacian matrix for the penta-chain network ('O). Furthermore, by utilizing the roots and coefficients of the obtained matrices, we derived formulas for both the mean first-passage time (MFPT) and the Kemeny's constant (KC) for'O. Finally, we compared the result of MFPT and KC with the number of pentagons.