The mesh matrix $Mesh(G,T_0)$ of a connected finite graph
$G=(V(G),E(G))=(vertices, edges) \ of \ G$ of with respect to a choice of a
spanning tree $T_0 \subset G$ is defined and studied. It was introduced by
Trent \cite{Trent1,Trent2}. Its characteristic polynomial $det(X \cdot Id
-Mesh(G,T_0))$ is shown to equal $\Sigma_{j=0}^{N} \ (-1)^j \ ST_{j}(G,T_0)\
(X-1)^{N-j} \ (\star)$\ where$ST_j(G,T_0)$is the number of spanning trees of$G$meeting$E(G-T_0)$in j edges and$N=|E(G-T_0)|$. As a consequence, there
are Tutte-type deletion-contraction formulae for computing this polynomial.
Additionally,$Mesh(G,T_0) -Id$is of the special form$Y^t \cdot Y$; so the
eigenvalues of the mesh matrix$Mesh(G,T_0)$are all real and are furthermore
be shown to be$\ge +1$. It is shown that$Y \cdot Y^t$, called the mesh
Laplacian, is a generalization of the standard graph Kirchhoff Laplacian$\Delta(H)= Deg -Adj$of a graph$H$.For example,$(\star)$generalizes the all
minors matrix tree theorem for graphs$H$and gives a deletion-contraction
formula for the characteristic polynomial of$\Delta(H)$. This generalization
is explored in some detail. The smallest positive eigenvalue of the mesh
Laplacian, a measure of flux, is estimated, thus extending the classical
inequality for the Kirchoff Laplacian of graphs.Comment: 21 Page