12 research outputs found
Graphs with few trivial characteristic ideals
We give a characterization of the graphs with at most three trivial characteristic ideals. This implies the complete characterization of the regular graphs whose critical groups have at most three invariant factors equal to 1 and the characterization of the graphs whose Smith groups have at most 3 invariant factors equal to 1. We also give an alternative and simpler way to obtain the characterization of the graphs whose Smith groups have at most 3 invariant factors equal to 1, and a list of minimal forbidden graphs for the family of graphs with Smith group having at most 4 invariant factors equal to 1
The degree-distance and transmission-adjacency matrices
Let be a connected graph with adjacency matrix . The distance
matrix of has rows and columns indexed by with -entry
equal to the distance which is the number of edges in a
shortest path between the vertices and . The transmission
of is defined as .
Let be the diagonal matrix with the transmissions of the
vertices of in the diagonal, and the diagonal matrix with
the degrees of the vertices in the diagonal. In this paper we investigate the
Smith normal form (SNF) and the spectrum of the matrices
,
,
and
. In particular, we explore how good
the spectrum and the SNF of these matrices are for determining graphs up to
isomorphism. We found that the SNF of has an interesting
behaviour when compared with other classical matrices. We note that the SNF of
can be used to compute the structure of the sandpile group
of certain graphs. We compute the SNF of ,
, and for several
graph families. We prove that complete graphs are determined by the SNF of
, , and
. Finally, we derive some results about the spectrum of
and .Comment: 19 page
Spectral integral variation of signed graphs
We characterize when the spectral variation of the signed Laplacian matrices
is integral after a new edge is added to a signed graph. As an application, for
every fixed signed complete graph, we fully characterize the class of signed
graphs to which one can recursively add new edges keeping spectral integral
variation to make the signed complete graph.Comment: 18 pages, 0 figur
Študija izvedljivosti - ekonomska upravičenost postavitve sončne elektrarne
summary:A graph is called distance integral (or -integral) if all eigenvalues of its distance matrix are integers. In their study of -integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on -integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs with , and with , as well as the infinite classes of distance integral complete multipartite graphs with