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Suppose G is a fixed but arbitrary graph. Prove or Disprove that the roots of p(G,x)...

Suppose G is a fixed but arbitrary graph. Prove or Disprove that the roots of p(G,x) are all real.

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Answer #1

Prove: All roots of p(G,x) are all real.

Proof: Let us first recall the definition of Characteristic polynomial p(G,x).

The characteristic polynomial of G is defined as

where is adjacency matrix of G.

No matter what is the graph G, the adjacency matrix of G is always is a symmetric matrix. That means, p(G,x) is nothing but a characteristic polynomial of a symmetric matrix. Now we know that the root of characteristic polynomial of a symmetrix matrx (also known as eigenvalue) is always real. Hence all roots of p(G,x) are real.

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