Signed Complete Graphs with Negative Paths

Authors

  • Soudabeh Dalvandi Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran
  • Farideh Heydari Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran
  • Mohammad Maghasedi Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

DOI:

https://doi.org/10.30495/jme.v15i0.1305

Keywords:

Signed graph, Complete graph, Path, Adjacency matrix

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 $x$.

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Published

2019-12-04

Issue

Vol. 15

Section

Vol. 15, No. 1, (2021)

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