What is "conductance"

Conductance \Con*duct"ance\ (k[o^]n*d[u^]k"tans), n. [Conduct, v. + -ance.] (Elec.) Conducting power; -- the reciprocal of resistance. A suggested unit is the mho, the reciprocal of the ohm.

Conductance is an attribute of any specified conductor, and refers to its shape, length, and other factors. Conductivity is an attribute of any specified material without direct reference to its shape or other factors.
--Sloane's Elec. Dict.

n. (context physics English) A measure of the ability of a body to conduct electricity; the reciprocal of its resistance.

n. a material's capacity to conduct electricity; measured as the reciprocal of electrical resistance

Conductance may refer to:

• Electrical conductance, the ability for electricity to flow a certain path
• Fluid conductance, the ability for fluid to transmit through materials
• Thermal conductivity, the ability for temperatures to transmit through materials
• Conductance (graph), a measure in graph theory

In graph theory the conductance of a graph G=(V,E) measures how "well-knit" the graph is: it controls how fast a random walk on G converges to a uniform distribution. The conductance of a graph is often called the Cheeger constant of a graph as the analog of its counterpart in spectral geometry. Since electrical networks are intimately related to random walks with a long history in the usage of the term "conductance", this alternative name helps avoid possible confusion.

The conductance of a cut $(S, \bar S)$ in a graph is defined as:

$$\varphi(S) = \frac{\sum_{i \in S, j \in \bar S}a_{ij}}{\min(a(S),a(\bar S))}$$

where the a are the entries of the adjacency matrix for G, so that

a(S) = ∑∑a

is the total number (or weight) of the edges incident with S.

The conductance of the whole graph is the minimum conductance over all the possible cuts:

ϕ(G) = minφ(S).

Equivalently, conductance of a graph is defined as follows:

$\phi(G) := \min_{S \subseteq V; 0\leq a(S)\leq a(V)/2}\frac{\sum_{i \in S, j \in \bar S}a_{ij}}{a(S)}.\,$

For a d-regular graph, the conductance is equal to the isoperimetric number divided by d.