Characterization of singular graphs can be reduced to the non-trivial solutions of a system of linear homogeneous equations Ax = 0 for the 0-1 adjacency matrix A. A graph G is singular of nullity η(G) ≥ 1, if the dimension of the nullspace ker(A) of its adjacency matrix A is η(G). Necessary and sufficient conditions are determined for a graph to be singular in terms of admissible induced subgra...

The Colin de Verdière parameters, μ and ν, are defined to be the maximum nullity of certain real symmetric matrices associated with a given graph. In this work, both of these parameters are calculated for all chordal graphs. For ν the calculation is based solely on maximal cliques, while for μ the calculation depends on split subgraphs. For the case of μ our work extends some recent work on com...

In this paper, we prove some refined estimate in the neck region when a sequence of harmonic maps from surfaces blow up. The new puts more restrictions to bubble and weak limit map than touching required by classical no-neck theorem. As an application, inequality about nullity index blow-up occurs.