On the nullity of bipartite graphs

Bipartite graphs with at most six non-zero eigenvalues

“…On the other hand, the main step to find a classification for triangle-free graphs of rank 6 is first to classify bipartite graphs of the same rank (see [9]). In this paper, in a different way from [5] and [9], we completely characterize all bipartite graphs with rank r, where r ∈ {2, 4, 6}. In other words, we find all bipartite graphs with at most six non-zero eigenvalues.…”

“…We note that by a different method in [5], the bipartite graphs of rank 4 are classified. for some real α.…”

On bipartite graphs which attain minimum rank among bipartite graphs with given diameter

“…Cheng and Liu [6] characterized the graphs of order n with nullity n − 2 or n − 3 or, equivalently, the graphs with rank 2 or 3. Fan and Qian [10] characterized all bipartite graphs with rank 4. Recently Chang, Huang, and Yeh characterized the graphs with rank 4 in [4] and also the graphs with rank 5 in [5].…”

On the algorithmic complexity of zero-sum edge-coloring

“…For more examples, see [4,6,8,11,15,20]. A zero-sum k-flow for a graph G is a vector in the null space of the 0,1-incidence matrix of G such that its entries belong to {±1, · · · , ±(k − 1)}.…”