A Note on Spectrum of Signed Complete Graphs

A Note on Spectrum of Signed Complete Graphs

Let Γ = (G,σ) beasignedgraph, where G istheunderlyingsimplegraphand σ : E(G) −→{−,+} is the sign function on the edges of G. The adjacency matrix of a signed graph has −1 or +1 for adjacent vertices, depending on the sign of the connecting edges. In this talk, we show that −1 and 1 are the eigenvalues of the signed complete graph with the multiplicity at least t if there are t + 1 vertices whose all incident edges are positive or negative, respectively. Also, we study the spectrum of a signed complete graph whose negative edges induce an r-regular subgraph H.

Let Γ = (G,σ) beasignedgraph, where G istheunderlyingsimplegraphand σ : E(G) −→{−,+} is the sign function on the edges of G. The adjacency matrix of a signed graph has −1 or +1 for adjacent vertices, depending on the sign of the connecting edges. In this talk, we show that −1 and 1 are the eigenvalues of the signed complete graph with the multiplicity at least t if there are t + 1 vertices whose all incident edges are positive or negative, respectively. Also, we study the spectrum of a signed complete graph whose negative edges induce an r-regular subgraph H.