Enhancement in energy recovery is always an essential element that requires academic spotlights to ensure its capability to contribute towards carbon neutrality. Recent works have extended to cover multi-solution heat exchanger networks (HEN) synthesis instead of generating a single best solution, which is not guaranteed to be practical. Nevertheless, owing to the technical challenges of synthesising all feasible networks, none of the existing works attempts to comprehensively elucidate how network topologies affect the network cost. To address this gap, P-HENS, a graph theoretic-based HEN synthesis tool, was utilised to generate the set of all heat exchanger networks with minimum utility consumption. Its effectiveness is demonstrated through an illustrative case study, which eventually generates more than 45,000 HENs. The impacts of structural variables on the cost, including the number of exchangers and the stream pairings, were analysed. The cost range of the networks was identified, revealing cost differences of 30 % despite minimum utility consumption or 15 % despite the minimum number of exchangers. Key stream pairs required to meet maximum energy recovery and influence cost were identified, leading to recommendations for improving solution searches. The solution set and the insight from this work are available to the research community for further analysis, offering valuable insights to enhance energy integration in the industry.