Ela Colin De Verdière Parameters of Chordal Graphs
نویسنده
چکیده
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 computing μ for split graphs.
منابع مشابه
Ela a Variant on the Graph Parameters of Colin De Verdière: Implications to the Minimum Rank of Graphs∗
For a given undirected graph G, the minimum rank of G is defined to be the smallest possible rank over all real symmetric matrices A whose (i, j)th entry is nonzero whenever i = j and {i, j} is an edge in G. Building upon recent work involving maximal coranks (or nullities) of certain symmetric matrices associated with a graph, a new parameter ξ is introduced that is based on the corank of a di...
متن کاملColin de Verdiere parameters of chordal graphs
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...
متن کاملThe Colin de Verdière parameter, excluded minors, and the spectral radius
In this paper we characterize graphs which maximize the spectral radius of their adjacency matrix over all graphs of Colin de Verdière parameter at most m. We also characterize graphs of maximum spectral radius with no H as a minor when H is either Kr or Ks,t. Interestingly, the extremal graphs match those which maximize the number of edges over all graphs with no H as a minor when r and s are ...
متن کاملThe Colin De Verdière Graph Parameter for Threshold Graphs
We consider Schrödinger operators on threshold graphs and give an explicit construction of a Colin de Verdière matrix for each connected threshold graph G of n vertices. We conclude the Colin de Verdière graph parameter μ(G) satisfies μ(G) ≥ n− i− 1, where i is the number of isolates in the graph building sequence. The proof is algorithmic in nature, constructing a particular Colin de Verdiére ...
متن کاملColin de Verdière number and graphs of polytopes
To every convex d-polytope with the dual graph G a matrix is associated. The matrix is shown to be a discrete Schrödinger operator on G with the second least eigenvalue of multiplicity d. This implies that the Colin de Verdière parameter of G is greater or equal d. The construction generalizes the one given by Lovász in the case d = 3.
متن کامل