This work elaborates on the old problem of measuring the degree of similarity, say \({\mathfrak {f}}(G,H)\), between a pair of connected graphs G and H, not necessarily of the same order. The choice of a similarity index \({\mathfrak {f}}\) depends essentially on the graph properties that are considered as important in a given context. As relevant information on a graph, one may consider for instance its degree sequence, its characteristic polynomial, and so on. We explore some new similarity indices based on nonstandard spectral information contained in the graphs under comparison. By nonstandard spectral information in a graph, we mean the set of complementarity eigenvalues of the adjacency matrix. From such a spectral perspective, two distinct graphs G and H are viewed as highly similar if they share a large number of complementarity eigenvalues. This basic idea will be cast in a rigorous mathematical formalism.

这项工作阐述了测量一对连接图G和H之间的相似度的老问题，例如\（{\ mathfrak {f}}（G，H）\），不一定是相同顺序的。相似度索引\（{\ mathfrak {f}} \）的选择基本上取决于在给定上下文中被视为重要的图形属性。作为图上的相关信息，可以考虑例如其度数序列，其特征多项式等。我们基于比较图中的非标准光谱信息探索了一些新的相似性指标。通过图中的非标准频谱信息，我们指的是邻接矩阵的互补特征值集。从这样的光谱角度来看，如果两个不同的图G和H共享大量的互补特征值，则它们被视为高度相似。这个基本思想将被应用在严格的数学形式主义中。