Relationship between the rank and the matching number of a graph

Mixed graphs with cut vertices having exactly two positive eigenvalues

“…The spectral-based graph invariants are widely investigated in the literature, such as inertia index [9,12,15,23,39,37,42,52,44,49,56,57,61] of simple graphs, signed graphs, mixed graphs and complex unit gain graphs, nullity [3,7,6,8,10,14,38,41,45,56,57,64] of simple graphs and signed graphs, rank [1,11,46,47,48,54] of simple graphs, signed graphs and complex unit gain graphs, the H-rank [4,5,50] of mixed graphs and skew-rank [20,22,32,33,26,27,40,53,62] of oriented graphs. The spectral parameter "inertia index" can determine the structure of a graph to some extent and has attracted much attention recently.…”

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.…”

Inertia indices of a complex unit gain graph in terms of matching number

“…Collatz et al [1] had wanted to obtain all graphs of order n with r(G) < n. Until today, this problem is also unsolved. In mathematics, the rank (or nullity, inertia index) of a graph has attracted a lot of researchers' attention, they focus on the bounds for the rank (or nullity, inertia index) of a simple graph G [4,6,11,23,24,25,26,34], a signed graph [3,5,8,15,19,32], an oriented graph [12,14,16,20,21,28] and a mixed graph [2,29] and so on.…”