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Introduction to Graph Theory pdf

Introduction to Graph Theory

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These notes are primarily a digression to provide general background remarks. The subject is an
efficient procedure for the determination of voltages and currents of a given network. A network
comprised of B branches involves 2B unknowns, i.e., each of the branch voltages and currents.
However the branch volt-ampere relations of the network, presumed to be known, relate the current and
the voltage of each branch,. Hence a calculation of either B currents or B voltages (or some
combination of B voltages and currents), and then substitution in the B branch volt-ampere relations,
provides all the voltages and currents.
In general however neither the B branch voltages nor the B branch currents are independent, i.e., some
of the B voltage variables for example can be expressed as a combination of other voltages using KVL,
and some of the branch currents can be related using KCL. Hence there generally are fewer than B
independent unknowns. In the following notes we determine the minimum number of independent
variables for a network analysis, the relationship between the independent and dependent variables, and
efficient methods of obtaining independent equations to determine the variables. In doing so we make
use of the mathematics of Graph Theory.

Graph Theory

A circuit graph is a description of the just the topology of the circuit, with details of the circuit elements
suppressed. The graph contains branches and nodes. A branch is a curve drawn between two nodes to
indicate an electrical connection between the nodes.
A directed graph is one for which a polarity marking is assigned
to all branches (usually an arrow) to distinguish between
movement from node A to B and the converse movement from
B to A.
A connected graph is one in which there is a continuous path
through all the branches (any of which may be traversed more
than once) which touches all the nodes. A graph that is not
connected in effect has completely separate parts, and for our
purposes is more conveniently considered to be two (or more)
independent graphs.

Choosing Independent Current Variables:

Given a network graph with B branches and N nodes select a tree, any one will do for the present
purpose. Remove all the link branches so that, by definition, there are no loops formed by the remaining
tree branches. It follows from the absence of any closed paths that all the branch currents become zero.
Hence by 'controlling' just the link currents all the branch currents can be controlled. This control would
not exist in general using fewer than all the link branches because a loop would be left over; depending
on the nature of the circuit elements branches making up the loop current could circulate around the
loop. Using more than the link branches is not necessary. Hence it should be possible to express all the
branch currents in terms of just the link currents, i.e., there are B-N+1 independent current variables, and
link currents provide one such set of independent variables.
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