A signed graph $\Gamma$ is a graph whose edges are labeled by signs. If $\Gamma$ has $n$ vertices, its spectral radius is the number $\rho \left(\Gamma \right):=max\left\{\right|{\lambda }_i\left(\Gamma \right)|:1\le i\le n\}$, where ${\lambda }_1\left(\Gamma \right)\ge \cdots \ge {\lambda }_n\left(\Gamma \right)$ are the eigenvalues of the signed adjacency matrix $A\left(\Gamma \right)$. Here we determine the signed graphs achieving the minimal or the maximal spectral radius in the classes ${𝔘}_n$ and ${𝔅}_n$ of unbalanced unicyclic graphs and unbalanced bicyclic graphs, respectively.

A graph is nonsingular if its adjacency matrix $A\left(G\right)$ is nonsingular. The inverse of a nonsingular graph $G$ is a graph whose adjacency matrix is similar to $A{\left(G\right)}^{-1}$ via a particular type of similarity. Let $\mathscr{H}$ denote the class of connected bipartite graphs with unique perfect matchings. Tifenbach and Kirkland (2009) characterized the unicyclic graphs in $\mathscr{H}$ which possess unicyclic inverses. We present a characterization of unicyclic graphs in $\mathscr{H}$ which possess bicyclic inverses.

One of the best known results in spectral graph theory is the following lower bound on the chromatic number due to Alan Hoffman, where μ1 and μn are respectively the maximum and minimum eigenvalues of the adjacency matrix: χ ≥ 1+μ1/−μn. We recently generalised this bound to include all eigenvalues of the adjacency matrix. In this paper, we further generalize these results to include all eigenvalues of the adjacency, Laplacian and signless Laplacian matrices. The various known bounds are also unified...

We present a lower and an upper bound for the second smallest eigenvalue of Laplacian matrices in terms of the averaged minimal cut of weighted graphs. This is used to obtain an upper bound for the real parts of the non-maximal eigenvalues of irreducible nonnegative matrices. The result can be applied to Markov chains.