In a signed graph $G$, an induced subgraph is called a negative clique if it is a complete graph and all of its edges are negative. In this paper, we give the characteristic polynomials and the eigenvalues of some signed graphs having negative cliques. This includes cycle graphs, path graphs, complete graphs with vertex-disjoint negative cliques of different orders, and star block graphs with negative cliques. Interestingly, if we reverse the signs of the edges of these graphs, we get the families of weakly balanced signed graphs, thus the eigenvalues of wide classes of weakly balanced signed graphs are also calculated. In social network theory, the eigenvalues of the signed graphs play an important role in determining their stability and developing the measures for the degree of balance.
Extension of the Gyárfás-Sumner conjecture to signed graphs
Université Paris-Panthéon-Assas（巴黎先贤祠-阿萨斯大学）；Universidade Federal do Ceará（塞阿拉联邦大学）；Université Paris Cité（巴黎西岱大学）；CNRS（法国国家科学研究中心）；IRIF（IRIF实验室）；Zhejiang Normal University（浙江师范大学）
Balanced-chromatic number and Hadwiger-like conjectures
Universidad de Valparaíso（瓦尔帕莱索大学）；Millennium Nucleus for Social Data Science (SODAS)（社会数据科学千年核（SODAS））；Auburn University（奥本大学）；Université Paris Cité（巴黎西岱大学）；CNRS（法国国家科学研究中心）；IRIF（信息研究与研究所）；École Normale Supérieure, Paris（巴黎高等师范学院）
