The stochastic gravitational wave background (SGWB) reveals valuable information about its origin and the Universe. The pulsar timing arrays (PTAs) are suitable indicators for detecting SGWB within the nano-Hertz frequency range. In this work, we propose a graph-based method implemented on the pulsar timing residuals (PTRs) for SGWB detection and examining uncertainties of its parameters. We construct a correlation graph with pulsars as its nodes, and analyze the graph-based summary statistics, which include topological and geometrical characteristics, for identifying SGWB in real and synthetic datasets. The effect of the number of pulsars, the observation time span, and the strength of the SGWB on the graph-based feature vector is evaluated. Our results demonstrate that the merit feature vector for common signal detection consists of the average clustering coefficient and the edge weight fluctuation. The SGWB detection conducted after the observation of a common signal and then exclusion of non-Hellings \& Downs templates is performed by the second cumulant of edge weight for angular separation thresholds $\bar{\zeta}\gtrsim 40^{\circ}$. The lowest detectable value of SGWB strain amplitude utilizing our graph-based measures at the current PTAs sensitivity is $A_{\rm SGWB}\gtrsim 1.2\times 10^{-15}$. Fisher forecasts confirmed that the uncertainty levels of $\log_{10} A_{\rm SGWB}$ and spectral index reach $2.2\%$ and $28.3\%$, respectively, at $2\sigma$ confidence interval. Evidence for an SGWB at the $3\sigma$ level is obtained by applying our graph-based method to the NANOGrav 15-year dataset.

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