This analysis uncovers singular conditions for tricyclic graphs through determinants of adjacency matrices, indicating diverse probabilities of singularity.
A graph G is said to be singular if its adjacency matrix is a singular matrix. In this paper, by analyzing the structure of Sachs subgraphs on three types of tricyclic graphs ϖ(a1,a2,b1,b2,c1,c2), σ(a1,a2,b1,b2,c), and τ(a1,a2,a3,a4) and calculating the determinants of their adjacency matrices, we investigate the necessary and sufficient conditions for the singularity of these three types of tricyclic graphs. Furthermore, we derive that the probabilities of singular graphs among tricyclic graphs ϖ(a1,a2,b1,b2,c1,c2), σ(a1,a2,b1,b2,c), and τ(a1,a2,a3,a4) are 217512,113256, and 67128, respectively.
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Ma et al. (2025) studied this question.