图核的定义是:

令G是一个有限的或无限的图的集合,函数k:G✘G->R称为一个图核,如果存在一个希尔伯特空间(可能无限维)F和一个映射φ :G→F ,使得对于所有的g ,g＇∈ G ,k(g ,g＇)=<φ(g),φ(g＇) ,<·,·>表示希尔伯特空间上的点积。根据这个定义,每个图核 k 可以看成希尔伯特空间 F 中的点积,换句话说,不需要定义从 G 到 F 的映射 ,只需要计算它们的点积,就能够简单地计算出核函数的值。由此可见，所谓图核，就是一个用于度量图相似性的核函数，把图映射到特征向量空间，使得两个图的相似性就是它们在特征向量空间的点积。

In structure mining, a domain of learning on structured data objects in machine learning, a graph kernel is a kernel function that computes an inner product on graphs.[1] Graph kernels can be intuitively understood as functions measuring the similarity of pairs of graphs. They allow kernelized learning algorithms such as support vector machines to work directly on graphs, without having to do feature extraction to transform them to fixed-length, real-valued feature vectors. They find applications in bioinformatics, in chemoinformatics (as a type of molecule kernels[2]), and in social network analysis.[1]

Concepts of graph kernels have been around since the 1999, when D. Haussler[3] introduced convolutional kernels on discrete structures. The term graph kernels was more officially coined in 2002 by R. I. Kondor and John Lafferty[4] as kernels on graphs, i.e. similarity functions between the nodes of a single graph, with the World Wide Web hyperlink graph as a suggested application. Vishwanathan et al. instead defined kernels between graphs.[1] In 2018, Ghosh et al. [5] described the history of graph kernels and their evolution over two decades.

An example of a kernel between graphs is the random walk kernel, which conceptually performs random walks on two graphs simultaneously, then counts the number of paths that were produced by both walks. This is equivalent to doing random walks on the direct product of the pair of graphs, and from this, a kernel can be derived that can be efficiently computed.[1]

References
[1] S.V. N. Vishwanathan; Nicol N. Schraudolph; Risi Kondor; Karsten M. Borgwardt (2010). "Graph kernels" (PDF). Journal of Machine Learning Research. 11: 1201–1242.
[2] L. Ralaivola; S. J. Swamidass; S. Hiroto; P. Baldi (2005). "Graph kernels for chemical informatics". Neural Networks. 18: 1093–1110. doi:10.1016/j.neunet.2005.07.009.
[3] Haussler, David (1999). Convolution Kernels on Discrete Structures.
[4]Risi Imre Kondor; John Lafferty (2002). Diffusion Kernels on Graphs and Other Discrete Input Spaces (PDF). Proc. Intl Conf. on Machine Learning (ICML).
[5]Ghosh, Swarnendu; Das, Nibaran; Gonçalves, Teresa; Quaresma, Paulo; Kundu, Mahantapas. "The journey of graph kernels through two decades". Computer Science Review. 27: 88–111. doi:10.1016/j.cosrev.2017.11.002.