Some classes of connected signed graphs with girth $g$ and negative inertia index $\lceil \frac{g}{2}\rceil +1$

Abstract

Let $\Gamma$ be a signed graph. The number of negative eigenvalues of the adjacency matrix of $\Gamma$ is called the negative inertia index of $\Gamma$, which is denoted by $i_{-}(\Gamma )$. The length of the shortest cycle contained in $\Gamma$ is called the girth of $\Gamma$, and it is denoted by $g$. In this paper, we give some classes of connected signed graphs $\Gamma$ which satisfy the condition $i_{-}(\Gamma )=\lceil \frac{g}{2}\rceil +1$.