Signed Complete Graphs with Negative Paths

Abstract

Let $\Gamma=(G,\sigma)$ be a signed graph, where $G$ is the underlying simple graph and $\sigma : E(G) \longrightarrow \lbrace -,+\rbrace$ 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 $\Gamma=(K_{n},\bigcup_{i=1}^{m}P_{r_i}^- )$ 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 _{i=1}^{m}\big(\lfloor \frac{r_i}{2}\rfloor-r_i\big) $ eigenvalues of $\Gamma$, where $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.