4 research outputs found
Relation between the skew-rank of an oriented graph and the independence number of its underlying graph
An oriented graph is a digraph without loops or multiple arcs
whose underlying graph is . Let be the
skew-adjacency matrix of and be the independence number
of . The rank of is called the skew-rank of ,
denoted by . Wong et al. [European J. Combin. 54 (2016) 76-86]
studied the relationship between the skew-rank of an oriented graph and the
rank of its underlying graph. In this paper, the correlation involving the
skew-rank, the independence number, and some other parameters are considered.
First we show that , where
is the order of and is the dimension of cycle space of .
We also obtain sharp lower bounds for , and characterize all
corresponding extremal graphs.Comment: 16 Page; 1 figur
Complex unit gain bicyclic graphs with rank 2, 3 or 4
A -gain graph is a triple
consisting of a graph , the circle group and a gain function
such that .
The rank of -gain graph , denoted by , is the rank
of the adjacency matrix of . In 2015, Yu, Qu and Tu [ G. H. Yu, H. Qu, J.
H. Tu, Inertia of complex unit gain graphs, Appl. Math. Comput. 265(2015)
619--629 ] obtained some properties of inertia of a -gain graph.
They characterized the -gain unicyclic graphs with small positive
or negative index. Motivated by above, in this paper, we characterize the
complex unit gain bicyclic graphs with rank 2, 3 or 4.Comment: 15 pages, 4 figure
The rank of a complex unit gain graph in terms of the rank of its underlying graph
Let be a complex unit gain graph (or -gain
graph) and be its adjacency matrix, where is called the
underlying graph of . The rank of , denoted by , is the
rank of . Denote by the dimension
of cycle spaces of , where , and are the number
of edges, the number of vertices and the number of connected components of ,
respectively. In this paper, we investigate bounds for in terms of
, that is, , where
is the rank of . As an application, we also prove that
. All corresponding
extremal graphs are characterized.Comment: 17 pages. arXiv admin note: text overlap with arXiv:1612.0504
Mixed graphs with cut vertices having exactly two positive eigenvalues
A mixed graph is obtained by orienting some edges of a simple graph. The
positive inertia index of a mixed graph is defined as the number of positive
eigenvalues of its Hermitian adjacency matrix, including multiplicities. This
matrix was introduced by Liu and Li, independently by Guo and Mohar, in the
study of graph energy. Recently, Yuan et al. characterized the mixed graphs
with exactly one positive eigenvalue. In this paper, we study the positive
inertia indices of mixed graphs and characterize the mixed graphs with cut
vertices having positive inertia index 2.Comment: 26 pages, 4 figure