## 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.

### MSC:

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

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