An introduction to analytic graph theory (Q2725009)

Suppose \(\mathcal S\) is a set of graphs or a set of objects associated with some specific graph such that there is a symmetric adjacency relation defined on \(\mathcal S\). Two elements \(S\) and \(S'\) are connected in \(\mathcal S\) if there is a sequence \(S=S_0, S_1, \ldots, S_k = S'\) of elements of \(\mathcal S\) such that \(S_i\) and \(S_{i+1}\) are adjacent for \(i = 0, 1, \ldots, k-1\). The minimum \(k\) for which such a sequence exists is called the distance between \(S\) and \(S'\) and is denoted by \(d(S, S')\). (For example, suppose \(G\) and \(H\) are graphs of orders \(n\) and \(k\), respectively, where \(n \geq k\). Then vertices \(u\) and \(v\) of \(G\) are said to be \(H\)-adjacent in \(G\) if they belong to the same copy of \(H\) in \(G\).) If \(\mathcal S\) is connected, then \((\mathcal S\), \( d)\) is a metric space. The authors define a continuity notion for integer valued functions \(f\) on \(\mathcal S\). They obtain results that resemble some well-known results for continuous real valued functions and they study specific functions that arise naturally from a variety of graph invariants.