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Multiplicity of integer roots of polynomials of graphs. (English) Zbl 0843.05074
Let $$G$$ be a graph and let $$\delta$$ be the minimal degree of vertices in $$G$$. Let $$B= D+ A$$, where $$D$$ is the diagonal matrix of vertex degrees and $$A$$ is the adjacency matrix of $$G$$. A combinatorial characterization is given for the multiplicity of $$\delta$$ as the root of the permanental polynomial $$\text{per}(xI- B)$$. If $$G$$ is bipartite, this characterization extends to $$\text{per}(xI- L)$$, where $$L= D- A$$ is the Laplacian matrix of $$G$$. These results are also extended to results about multiplicities of (arbitrary) integer roots of the permanental and the characteristic polynomials of both $$B$$ and $$L$$.

##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 15A15 Determinants, permanents, traces, other special matrix functions
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##### References:
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