| Summary: | The search for complete graph invariants is an important problem in graph theory and computer science. Two networks with a different structure can be distinguished from each other by complete graph invariants. In order to find a complete graph invariant, we introduce the generalized permanental polynomials of graphs. Let <i>G</i> be a graph with adjacency matrix <inline-formula> <math display="inline"> <semantics> <mrow> <mi>A</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and degree matrix <inline-formula> <math display="inline"> <semantics> <mrow> <mi>D</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. The generalized permanental polynomial of <i>G</i> is defined by <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>P</mi> <mi>G</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>μ</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>per</mi> <mrow> <mo>(</mo> <mi>x</mi> <mi>I</mi> <mo>−</mo> <mrow> <mo>(</mo> <mi>A</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>−</mo> <mi>μ</mi> <mi>D</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. In this paper, we compute the generalized permanental polynomials for all graphs on at most 10 vertices, and we count the numbers of such graphs for which there is another graph with the same generalized permanental polynomial. The present data show that the generalized permanental polynomial is quite efficient for distinguishing graphs. Furthermore, we can write <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>P</mi> <mi>G</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>μ</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> in the coefficient form <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>n</mi> </msubsup> <msub> <mi>c</mi> <mrow> <mi>μ</mi> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <msup> <mi>x</mi> <mrow> <mi>n</mi> <mo>−</mo> <mi>i</mi> </mrow> </msup> </mrow> </semantics> </math> </inline-formula> and obtain the combinatorial expressions for the first five coefficients <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>c</mi> <mrow> <mi>μ</mi> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mn>4</mn> </mrow> </semantics> </math> </inline-formula>) of <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>P</mi> <mi>G</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>μ</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>.
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