Signed Complete Graphs with Negative Paths

Let Gamma = (G, sigma) be a signed graph, where G is the underlying simple graph and sigma : 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. Let Gamma = (K-n, boolean OR(m)(i=1) P-ri(-)) be a signed complete graph whose negative edges induce a subgraph which is the disjoint union of m distinct paths. In this paper, by a constructive method, we obtain n -1 + Sigma(m)(i=1) (left perpendicular r(i)/2 right perpendicular - r(i)) eigenvalues of Gamma, where left perpendicular x right perpendicular denotes the largest integer less than or equal to x.