Inverse monoids of partial graph automorphisms

A partial automorphism of a finite graph is an isomorphism between its vertex induced subgraphs. The set of all partial automorphisms of a given finite graph forms an inverse monoid under composition (of partial maps). We describe the algebraic structure of such inverse monoids by the means of the standard tools of inverse semigroup theory, namely Green's relations and some properties of the natural partial order, and give a characterization of inverse monoids which arise as inverse monoids of partial graph automorphisms. We extend our results to digraphs and edge-colored digraphs as well

On the edge-reconstruction number of a tree

The edge-reconstruction number ern(G) of a graph G is equal to the minimum number of edge-deleted subgraphs G−e of G which are sufficient to determine G up to isomorphism. Building upon the work of Molina and using results from computer searches by Rivshin and more recent ones which we carried out, we show that, apart from three known exceptions, all bicentroidal trees have edge-reconstruction number equal to 2. We also exhibit the known trees having edge-reconstruction number equal to 3 and we conjecture that the three infinite families of unicentroidal trees which we have found to have edge-reconstruction number equal to 3 are the only ones.peer-reviewe

A note on vertex-transitive non-Cayley graphs from Cayley graphs generated by involutions

AbstractWe show that the result of Watkins (1990) [19] on constructing vertex-transitive non-Cayley graphs from line graphs yields a simple method that produces infinite families of vertex-transitive non-Cayley graphs from Cayley graphs generated by involutions. We also prove that the graphs arising this way are hamiltonian provided that their valency is at least six

Constructing Cospectral and Comatching Graphs

The matching polynomial is a graph polynomial that does not only have interesting mathematical properties, but also possesses meaningful applications in physics and chemistry. For a simple graph, the matching polynomial enumerates the number of matchings of different sizes in it. Two graphs are comatching if they have the same matching polynomial. Two vertices u, v in a graph G are comatching if G\ u and G\ v are comatching. In 1973, Schwenk proved almost every tree has the same characteristic polynomial with another tree. In this thesis, we extend Schwenk's result to maximal limbs and weighted trees. We also give a construction using 1-vertex extensions for comatching graphs and graphs with an arbitrarily large number of comatching vertices. In addition, we use an alternative definition of matching polynomial for multigraphs to derive new identities for the matching polynomial. These identities are tools used towards our 2-sum construction for comatching vertices and comatching graphs