To each weighted graph
Γ
\Gamma
, two invariants, a polynomial
P
Γ
(
x
,
y
,
z
)
{P_\Gamma }(x,y,z)
and the signature
σ
(
Γ
)
\sigma (\Gamma )
, are defined. The various partial degress of
P
Γ
(
x
,
y
,
z
)
{P_\Gamma }(x,y,z)
and
σ
(
Γ
)
\sigma (\Gamma )
are expressed in terms of maximal spanning graphs of
Γ
\Gamma
. Furthermore, one unexpected property of Tutte’s dichromate is proved. These results are applied to knots or links in
S
3
{S^3}
.