Tag Archives: Graph Theory

Group valued flows: A possible example

@IMSc, Chennai

Am here as an attendee for the NBHM nurture programme. Today, we had a talk by one of the participants, ADG on flows in graphs (ref. Diestel’s Graph Theory). In his talk, he defined the concept of group valued flows. Let there be a function f: \vec{E} \rightarrow H, where \vec{E} is the edge set of the multigraph G(V,\vec{E}) and H is an abelian group. Hence, f assigns to each edge of the edgeset \vec{E}, an element in H. This concept is an extension of the concept of integral flow in graphs, with H = \mathbb{Z} in the latter. I won’t go much into the theory itself, but I have some different objectives here.

After the lecture, one of the attendees complained that this notion of group valued flow is too abstract. So, a possible non-trivial example happened to occur to me in which the abelian group H doesn’t equal \mathbb{Z} \text{ or } \mathbb{R}. Consider, a hypothetical mode of communication (I don’t know if one such already exists), in which information happen to travel in the form of nuclei spin in the medium carrying it. So, the source or the transmitter orients the nuclei of the medium those are just adjacent to it, in a particular direction and the fraction of nuclei in that cross section which are oriented in that particular direction determine the magnitude of the spin vector. These nuclei then transfer this spin to their neighbours and reorient themselves randomly in all the directions such that their net spin vector again becomes zero. In this fashion, the information about the nuclei spin can flow through the medium. This flow can hence be thought of as a vector valued flow in which flow in each edge of the graph is given by a vector v \in \mathbb{R}^n. Hence, the group H happens to be the vector space \mathbb{R}^n having the usual addition operation of the vector space as its abelian operation.

P.S. 1) I’ve written this post primarily so that I can refer back to this idea whenever I might need it at some future point of time.

2) This is the first post belonging to the category math on my blog :) .

3) To write equations and other stuff in \LaTeX in your WordPress blog posts, type $latex yourcodehere $. Though I feel that WordPress doesn’t adjust the sentences involving LaTeX that good, but I have no other option ;) .