The extremal problems on the inertia of weighted bicyclic graphs

Let $G_w$ be a weighted graph. The number of the positive, negative and zero eigenvalues in the spectrum of $G_w$ are called positive inertia index, negative inertia index and nullity of $G_w$, and denoted by $i_{+}(G_w)$, $i_{-}(G_w)$, $i_{0}(G_w)$, respectively. In this paper, sharp lower bound on the positive (resp. negative) inertia index of weighted bicyclic graphs of order $n$ with pendant vertices is obtained. Moreover, all the weighted bicyclic graphs of order $n$ with at most two positive, two negative and at least $n-4$ zero eigenvalues are identified, respectively.Comment: 12 pages, 5 figures, 2 tables. arXiv admin note: text overlap with arXiv:1307.0059 by other author

A short proof of the odd-girth theorem

Recently, it has been shown that a connected graph $\Gamma$ with $d+1$ distinct eigenvalues and odd-girth $2d+1$ is distance-regular. The proof of this result was based on the spectral excess theorem. In this note we present an alternative and more direct proof which does not rely on the spectral excess theorem, but on a known characterization of distance-regular graphs in terms of the predistance polynomial of degree $d$

Further results on the nullity of signed graphs

The nullity of a graph is the multiplicity of the eigenvalues zero in its spectrum. A signed graph is a graph with a sign attached to each of its edges. In this paper, we obtain the coefficient theorem of the characteristic polynomial of a signed graph, give two formulae on the nullity of signed graphs with cut-points. As applications of the above results, we investigate the nullity of the bicyclic signed graph $\Gamma(\infty(p,q,l))$, obtain the nullity set of unbalanced bicyclic signed graphs, and thus determine the nullity set of bicyclic signed graphs.Comment: 17 pages. arXiv admin note: text overlap with arXiv:1207.6765, arXiv:1107.0400 by other author