minimum semidefinite rank, matrix of a graph, independence number, graph complement conjecture
The minimum semi-definite rank (msr) of a graph is the minimum rank among all positive semi-definite matrices associated to the graph. The graph complement conjecture gives an upper bound for the sum of the msr of a graph and the msr of its complement. It is shown that when the msr of a graph is equal to its independence number, the graph complement conjecture holds with a better upper bound. Several sufficient conditions are provided for the msr of different classes of graphs to equal to its independence number.
Narayan, Sivaram and Sharawi, Yousra.
"BOUNDS ON THE SUM OF MINIMUM SEMIDEFINITE RANK OF A GRAPH AND ITS COMPLEMENT",
Electronic Journal of Linear Algebra,
Volume 34, pp. 399-406.
DOI: https://doi.org/10.13001/1081-3810, 1537-9582.3539