Nullity of a graph in terms of the dimension of cycle space and the number of pendant vertices

A lower bound of the rank of a signed graph in terms of order and maximum degree

“…Collatz et al [1] attempted to obtain all graphs of order n with r(G) < n. Until today, this problem is still unsolved. In mathematics, the rank (or nullity) of a graph attracted a lot of researchers' attention, they focus on the relationship between the rank (or nullity) and some graph parameters, such as pendant vertices [2,3], matching number [4][5][6][7], path cover number [8], and so on.…”

Proof of a conjecture on the nullity of a graph

“…In 1957, Collatz and Sinogowitz [7] posed the problem of characterizing all singular graphs. The problem has received considerable attention in the literature (see, eg, [3‐5,10‐12,14‐21,24,26‐31,33] and references therein). In studying the above problem, some attentions are attracted to bound the nullity of a graph by using some of the structure parameters, such as the order, the matching number, the number of pendant vertices, and especially the maximum degree of the graph (see [5,6,8,12,14,15,17,22‐24,29,30] for details).…”

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

“…Chen et al [3] investigated the relation between the H-rank of a mixed graph and the matching number of its underlying graph. For other research of the rank of a graph one may be referred to those in [1,9,14,15,18].…”