Zoran Stanic: Some bounds for the largest eigenvalue of a signed graph, 183-189

Abstract:

For a vertex i of a signed graph, let $d_i$, $m_i$ and $T_i^-$ denote its degree, average 2-degree and the number of unbalanced triangles passing through i, respectively. We prove that

$$\begin{eqnarray*}\rho\leq\max\left\{ \dfrac{-d_i+\sqrt{5d_i^2+4(d_im_i-4T_i^-)}}{2}~:~ 1\leq i\leq n\right\}, \end{eqnarray*}$$

where $\rho$ stands for the largest eigenvalue. This bound is tight at least for regular signed graphs that are switching equivalent to their underlying graphs. We also derive a general lower bound for $\rho$ and certain practical consequences. A discussion, including the cases of equality in inequalities obtained and some examples, is given.

Key Words: Signed graph, switching equivalence, index, vertex degree, net-balance.

2010 Mathematics Subject Classification: Primary 05C50; Secondary 05C22.