Bounds for the matching number, the edge chromatic number and the independence number of a graph in terms of rank

An upper bound for the nullity of a bipartite graph in terms of its maximum degree

“…The aim of [5,6,11,16,17,20] is to bound the nullity of a class of graphs in terms of their orders. More recently, the nullity of an arbitrary graph was bounded in [19] in terms of matching number.…”

“…In 1957, Collatz and Sinogowitz [3] posed the problem of characterizing all singular graphs. The problem is very hard and only some particular results are known (see [2,[4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]). When studying above problem, some authors focused on describing the nullity set of certain class of graphs in terms of the order of graphs.…”

The rank of a complex unit gain graph in terms of the matching number

“…It is obvious that an undirected graph G is just a complex unit gain graph Φ = (G, T, ϕ) with ϕ( − → E ) ⊆ {1}. In [18], Wang et al researched the the relation among the rank, the matching number and the cyclomatic number of an undirected graph. Let T G be the graph obtained from G by contracting each cycle of G into a vertex (called a cyclic vertex).…”

“…Let T G be the graph obtained from G by contracting each cycle of G into a vertex (called a cyclic vertex). Wang et al [18] also characterized the undirected graphs G for which the lower bound and the upper bound for r(G) in Theorem 1.1 are obtained.…”

“…There are some papers focused on the study on the rank of graphs in terms of other topological structure parameters. Wang and Wong characterized the bounds for the matching number, the edge chromatic number and the independence number of a graph in terms of rank in [17]. Gutman and Sciriha [5] studied the nullity of line graphs of trees.…”