圖的特徵多項式與配對多項式

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weng@math.nctu.edu.tw

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Abstract Let G be a simple graph. It is known that when the graph G is a forest, the matching poly-nomial µ(G, x) is equal to the characteristic polyno-mial ϕ(G, x). But this is not true in general. In this project, we give a formula for ϕ(G, x) in terms of

µ(H, x) for some subgraphs H of G.

Keywords: characteristic polynomial; matching poly-nomial. Þ`ãêÝ ƒ G Î×Í n FÝ% G îÝ r-gEÎ×àÍ ó r Ý\/vP¢ËÍh/)Ý\F & Æà p(G, r) ¼‚h\/Ý-ôÍó p(G, 0) !  1 G ÝgE94PL µ(G, x) :=X r≥0 (−1)rp(G, r)xn−2r. ¨×]«% G Ý©Ç94PÎ ϕ(G, x) := det(xI − A(G)),

h A(G) Î% G Ý adjacency matrix @~gE 94PÎ×Íà)P²Ý"D
‚@~©Ç94P f´#‚óP²Ýá@
Ëïøݺà΂ó à).@~Ý]¶©‘ Gß@~á3% G ‘ݵì
% G ÝgE94Pª‡y©Ç94 P[1, p21] &ƘñµhÁìØTÿÕ?×M ݔŒ 딌D¡ AŒ G îb×Í\ e = {i, j} ‚v©b°××Í cycle C ‘â e
&ÆÿÕ ϕ(G, x) = ϕ(G \ e, x) − ϕ(G \ ij, x) − 2ϕ(G \ C, x). ×Í©½Î G b×Í cycle C v e = {i, j}Î C îÝØ\
&ÆÿÕ ϕ(G, x) = µ(G \ e, x) − µ(G \ ij, x) − 2µ(G \ C, x). 9ÍPÎÞ‚ó8n G Ý©Ç94Pàà )8n G Ý%ÝgE94P¼î &ÆTÞ ¼E G Ýf?wêôÿÕ?ݔŒ °ŒWŒŠÝ hŒi4ÿÕ×̛”¡
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1 D. Godsil, Algebraic Combinatorics, Chapman & Hall, Inc. 1993