"A graph is a simple and quite old mathematical concept: a data structure consisting of a set of vertices (or nodes/points) and edges (or relationships/lines) that can be used to model relationships among a collection of objects." (Alessandro Negro, "Graph-Powered Machine Learning", 2021)
"Graphs are useful for representing how things are either physically or logically linked in simple or complex structures. A graph in which we assign names and meanings to the edges and vertices becomes what is known as a network. In these cases, a graph is the mathematical model for describing a network, whereas a network is a set of relations between objects, which could include people, organizations, nations, items found in a Google search, brain cells, or electrical transformers." (Alessandro Negro, "Graph-Powered Machine Learning", 2021)
"Several areas of graph theory are concerned with the likelihood or certainty of the presence in a graph of various subgraphs or, more generally, of graph properties that emerge as the number of vertices and/or the number of edges increases. Collectively they are grouped as analytic graph theory." (Jonathan L Gross et al, "Topics in Graph Theory", 2023)
"Some connected graphs are 'more connected' than others. That is, a connected graph’s vulnerability to disconnection by edge- or vertex-deletion varies. Two numerical parameters, vertex-connectivity and edge-connectivity, are useful in measuring a graph’s connectedness. Intuitively, a network’s vulnerability should be closely related also to the number of alternative paths between each pair of nodes. There is a rich body of mathematical results concerning this relationship, many of which are variations of a classical result of Menger, and some of these extend well beyond graph theory." (Jonathan L Gross et al, "Topics in Graph Theory", 2023)
"Spanning trees capture the connectedness of a graph in the most efficient way, and they provide a foundation for a systematic analysis of the cycle structure of a graph. Mathematicians regard the algebraic structures underlying the collection of cycles and edge-cuts of agraph as beautiful in their own right. Establishing connections between linear algebra and graph theory provides some powerful analytical tools for understanding a graph’s structure." (Jonathan L Gross et al, "Topics in Graph Theory", 2023)
"The concept of temporal graphs is useful in all the real-world problems that can be represented as a graph, where the nodes and edges of the graph may change over time. For example, temporal graphs are extensively applied in modeling social networks. By capturing the evolving relationships between individuals, temporal graphs enable a more accurate representation of social dynamics. This is particularly useful for predicting changes in friendships, community structures, and the information diffusion over time." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
"Hierarchical clustering is summarised by a dendrogram, which sequentially shows points being joined to form a cluster, with the corresponding distances. Breaking the data into clusters is done by cutting the dendrogram at the long edges. [...] Plotting the dendrogram in the data space can help you understand how the hierarchical clustering has collected the points together into clusters. You can learn if the algorithm has been confused by nuisance patterns in the data, and how different choices of linkage method affect the result." (Dianne Cook & Ursula Laa, "Interactively Exploring High-Dimensional Data and Models in R", 2026)
"Viewing the dendrograms in high dimensions provides insight into how the algorithm has joined points to clusters. For example, single linkage often has edges leading to a single focal point, which might not yield a useful clustering but might help to identify outliers. If the edges point to multiple focal points, with long edges bridging gaps in the data, the result is more likely yielding a useful clustering." (Dianne Cook & Ursula Laa, "Interactively Exploring High-Dimensional Data and Models in R", 2026)
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