Definition. Weighted graph differentials and adjoints [fdip-0007]
AGENTDRAFTED
Let \(R\) be a commutative unital ring, \(V,E\) finite sets and
\(s,h:E\to V\) endpoint maps. The
ring-valued incidence map is
\(d_R:H_V\to H_E\), where \(H_V=\mathcal F_R(V)\),
\(H_E=\mathcal F_R(E)\) and \(d_Rf(e)=f(h(e))-f(s(e))\).
Equip both modules with the symmetric counting forms
\[b_V(f,g)=\sum _v f(v)g(v),\qquad b_E(a,c)=\sum _e a(e)c(e).\]
Their perfect counting dualities give the
counting adjoint \(d_R^\dagger :H_E\to H_V\) by
Proposition [fdip-001P]. It is characterized by
\(b_E(d_Rf,a)=b_V(f,d_R^\dagger a)\).
The return map can be read directly from the endpoints:
\[(d_R^\dagger a)(v)
=\sum _{h(e)=v}a(e)-\sum _{s(e)=v}a(e).\]
For arbitrary weights \(w:E\to R\), let
\(W:H_E\to H_E\) multiply pointwise by \(w\):
\((Wa)(e)=w(e)a(e)\). The weighted graph Laplacian is
\[L=d_R^\dagger Wd_R:H_V\longrightarrow H_V.\]
Commutativity makes \(W\) self-adjoint, since
\(b_E(Wa,c)=\sum _e w(e)a(e)c(e)=b_E(a,Wc)\).
The weighted edge form
\[b_{E,w}(a,c)=b_E(a,Wc)=\sum _e w(e)a(e)c(e)\]
is symmetric bilinear. It is perfect if and only if every \(w(e)\)
is a unit, meaning an invertible element of \(R\).
Even when some weights are nonunits, the represented return map
\(d_R^\dagger W\) exists and satisfies
\[b_{E,w}(d_Rf,a)=b_V(f,d_R^\dagger Wa).\]
Its existence does not make the weighted edge form perfect. Thus it
does not by itself give a two-sided adjoint involution for that form.
All these ring identities are direct deductions from the chosen
counting dualities. Their positive-real incidence convention is
[spielman2009effective, Section 2.1, p. 4].
Orientation reversal leaves \(L\) unchanged. Choose which edges to
reverse, and put \(\varepsilon (e)=-1\) on those edges and \(1\) on the
others, interpreted in \(R\). The sign map \(Ja(e)=\varepsilon (e)a(e)\)
satisfies \(J^\dagger =J\), \(J^2=I\), \(JW=WJ\), and the reversed
incidence map is \(d'_R=Jd_R\). Therefore
\[(d'_R)^\dagger Wd'_R=d_R^\dagger JWJd_R=d_R^\dagger Wd_R.\]
The corresponding edge field is \(a'=Ja\), and
\(b_{E,w}(Ja,Jc)=b_{E,w}(a,c)\). The proof includes characteristic two,
loops and parallel edges. Endpoints and weights are otherwise fixed.
For the real incidence adapter, start with the
incidence map
\(D:\mathbb R^V\to \mathbb R^E\) and positive weights \(w(e)>0\).
The counting forms are the counting inner products,
and write \(D^*=d_{\mathbb R}^\dagger \). The edge form becomes the
weighted inner product
\[\langle a,c\rangle _{E,w}=\sum _e w(e)a(e)c(e).\]
It is symmetric bilinear, and its self-pairing is a finite sum of
nonnegative terms. Strictly positive weights make that sum zero only
at the zero edge field. Keeping the vertex pairing counting, the
weighted adjoint is
\(D_w^\dagger =D^*W\), since
\[\langle Df,a\rangle _{E,w}
=\langle Df,Wa\rangle _E=\langle f,D^*Wa\rangle _V.\]
Hence \(L=D^*WD=D_w^\dagger D\); \(D^*D\) is the unit-weight case.
For the coordinate recovery, use the ordered-value isomorphisms
\(c_V,c_E\) and incidence matrix \(B=c_EDc_V^{-1}\).
The endpoint return formula gives
\(c_VD^*c_E^{-1}=B^T\). The multiplication map \(W\) has diagonal
array \(\operatorname {diag}(w(e))\); writing the same letter for that
array yields \([D_w^\dagger ]=B^TW\) and \([L]=B^TWB\).
For a single edge \(0\to 1\) of weight \(c>0\), the two adjoints send
an edge value \(a\) to \((-a,a)\) and \((-ca,ca)\), respectively.
Under orientation reversal the sign array \(S\) gives \(B'=SB\),
\(S^T=S\), \(S^2=I\), \(SWS=W\), and
\((B')^TWB'=B^TWB\). A loop has a zero incidence row before and
after reversal.
Mathlib's LinearMap.toMatrix_adjoint identifies adjoint
matrices with conjugate transposes in orthonormal bases. Over the reals
this is transpose. The coordinate edge basis is orthonormal for counting,
but not generally for the weighted pairing; the return-map calculation
above accounts for the extra \(W\).