Characteristic vertices of trees. (English) Zbl 0636.05021

Let \(G=(V,E)\) be a graph with vertex set \(V=\{v_ 1,v_ 2,...,v_ n\}\) and edge set E. Denote by L(G) the n-by-n-matrix \((a_{ij})\), where \(a_{ij}\) is the degree of vertex i when \(j=i\); \(a_{ij}=-1\) when \(j\neq i\) and \(\{\) i,j\(\}\in E\); and \(a_{ij}=0\), otherwise. While L(G) depends on the labeling of V, its characteristic polynomial \(q_ G(x)\) does not. The main result of the first section is a family of inequalities between the coefficients of \(q_ G(x)\) and the coefficients of the chromatic polynomial of G. If \(\lambda_ n\geq \lambda_{n-1}\geq...\geq \lambda_ 1\) are the eigenvalues of L(G), then \(\lambda_ 1=0\) and \(\lambda_ 2>0\) if and only if G is connected. For connected graphs, the eigenvectors of L(G) corresponding to \(\lambda_ 2\) afford “characteristic valuations” of G, a concept introduced by M. Fiedler. Sections II and III explore the “characteristic vertices” arising from characteristic valuations when G is a tree.


05C05 Trees
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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