The paper proposes a novel approach to bring out the potential of complex networks based on graph theory to unwrap the hidden characteristics of cough signals, croup (BC), and pertussis (PS). The spectral and complex network analyses of 48 cough sounds are utilized for understanding the airflow through the infected respiratory tract. Among the different phases of the cough sound time-domain signals of BC and PS – expulsive (X), intermediate (I), and voiced (V) - the phase ‘I’ is noisy in BC due to improper glottal functioning. The spectral analyses reveal high-frequency components in both cough signals with an additional high-intense low-frequency spread in BC. The complex network features created by the correlation mapping approach, like number of edges (E), graph density (G), transitivity (), degree centrality (D), average path length (L), and number of components () distinguishes BC and PS. The higher values of E, G, and for BC indicate its musical nature through the strong correlation between the signal segments and the presence of high-intense low-frequency components in BC, unlike that in PS. The values of D, L, and discriminate BC and PS in terms of the strength of the correlation between the nodes within them. The linear discriminant analysis (LDA) and quadratic support vector machine (QSVM) classifies BC and PS, with greater accuracy of 94.11% for LDA. The proposed work opens up the potentiality of employing complex networks for cough sound analysis, which is vital in the current scenario of COVID-19.