Growing odd graphs and asymptotic distribution in a deformed vacuum state

We investigate the asymptotic distribution of odd graphs in a deformed vacuum state, focusing on the spectral analysis of these graphs. We explore the adjacency matrices of odd graphs and derive explicit expressions for their mean and variance in the deformed vacuum state. Our main results provide the probability measures and the corresponding coherent states for the distribution of these graphs. We calculate the Jacobi coefficients and Cauchy transforms related to these distributions, which have not been addressed explicitly in the existing literature. Our findings contribute to a deeper understanding of the probabilistic and spectral properties of odd graphs in quantum state frameworks.