Consider G = (V, E) as a finite graph, where V and E correspond to the vertices and edges, respectively. We study a generalized Chern–Simons equation Δu=λeu(ebu1)+4πj=1Nδpj on G, where λ and b are positive constants; N is a positive integer; p1, p2, …, pN are distinct vertices of V; and δpj is the Dirac delta mass at pj. We prove that there exists a critical value λc such that the equation has a solution if λλc and the equation has no solution if λ < λc. We also prove that if λ > λc, the equation has at least two solutions that include a local minimizer for the corresponding functional and a mountain-pass type solution. Our results extend and complete those of Huang et al. [Commun. Math. Phys. 377(1), 613–621 (2020)] and Hou and Sun [Calculus Var. Partial Differ. Equations 61(4), 139 (2022)].

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