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

Abstract

Hui-xian. (2012), "On bipartite graphs which attain minimum rank among bipartite graphs with given diameter", Abstract. The rank of a graph is defined to be the rank of its adjacency matrix. In this paper, the bipartite graphs that attain the minimum rank among bipartite graphs with a given diameter are completely characterized.

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).…”

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

“…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.…”