The Minimal Positive Index of Inertia of Signed Unicyclic Graphs

Abstract

The positive index of inertia of a signed graph $\mathrm{\Gamma }$, denoted by $,(\mathrm{\Gamma })$, is the number of positive eigenvalues of the adjacency matrix $A(\mathrm{\Gamma })$ including multiplicities. In this paper we investigate the minimal positive index of inertia of
signed unicyclic graphs of order $n$ with fixed girth and characterize the extremal graphs with the minimal positive index. Finally, we characterize the signed unicyclic graphs with the positive indices $1$ and $2$.