If a graph represent a road network the height could represents the length of each road.There may be different heights assocoated with edge including distance,traveltimes or monetory cost. Such a weighted graphs are commonly used to program GPS and travel planning,search engine that compaces flight times and cost.
It is also used in mathematics as it is useful in geometry and certain part of topology.
It is useful in Biology too.Where vertex can represent certain spacies and egde represent migration paths or movement between species.Thir information is important when looking at the breeding patterns or tracking the spread.A diseases,parasites and how changes in the movement can affect other spacies too.
Graph theory is also used in physics and chemistry too.
In condensed matter physics ,three dimensional structure can be studided quantitatively by gathering statistics on graph theorethic properties.Chemical graph theory uses the molecular phase
as means of model moleculer.Graphs and network are excellent model to study and understand phase transition and critical phenomena.
Removal of nodes leads to critical transition where network breaks into small clusters which is studied as a phase transition.
Graph theory is also used in Computer sciences .It is used to represent networks of commonications,dataorganization,computational devices ,the flow of computation etc .For example the line strecture of a website can be represent webpages and directed edges lines from one page to another.