# Glossary: Spectra of matrices: notation

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 . The principal submatrix of whose rows and columns are indexed by (where .) . The vertex independence number of , i.e., the largest number for which a coclique with vertices exists. If is a loop-tree and , is the number of components of that allow eigenvalue . Cycle on vertices. The chromatic number of (provided does not have loops), i.e., the smallest number of color classes of any vertex coloring of . : there is a set of vertices whose deletion leaves paths.} (Note that an isolated vertex is a path of order 1 and is simple.) The diameter of , i.e., the maximum distance between any two vertices of G. The energy of the simple graph .     any field ( simple.) The simple graph of symmetric matrix , i.e., the simple graph with vertices and edges     and . Note that the diagonal of is ignored in determining . The result of deleting all vertices of and their incident edges from . The result of deleting a vertex of and its incident edges. The complement of simple graph , i.e., . The subgraph induced by , the subgraph with vertex set and edge set . The order of . The number of vertices in . The simple graph obtained from by removing all loops from . A graph, usually on . is the set of vertices, is the set of edges. is allowed to have loops and/or multiple edges. The geometric multiplicity of as an eigenvalue of (i.e., the dimension of ker( )). The set of high degree vertices of simple graph , i.e., the set of vertices of degree at least 3.     is a Hermitian     matrix over     and ( is simple.)     is positive semidefinite ( is simple.) ( is simple.) The complete (simple) graph on vertices. The complete (simple) bipartite graph on and vertices. The maximum multiplicity of simple , i.e., ( simple.) ( simple.) (It is necessary to distinguish between zero from nonzero eigenvalues because translation is no longer possible.) The minimum rank of simple graph , i.e., ( is simple.) ( is simple.) The Colin de Verdière number of simple graph is the maximum multiplicity of 0 as an eigenvalue among matrices that satisfy: is a generalized Laplacian matrix of . has exactly one negative eigenvalue (of multiplicity 1). satisfies the Strong Arnold Hypothesis. The multiplicity of as a root of the characteristic polynomial of (i.e., the algebraic multiplicity of if is an eigenvalue of and 0 otherwise). The rank-spread of simple graph at vertex is . The rank-spread of simple graph at vertex is . The parameter of simple graph is defined to be the maximum multiplicity of 0 as an eigenvalue among matrices that satisfy: . is positive semidefinite. satisfies the Strong Arnold Hypothesis. The parameter is defined to be the maximum multiplicity of 0 as an eigenvalue among matrices that satisfy: . satisfies the complex Strong Arnold Hypothesis. Same as The path cover number of simple graph , i.e., the minimum number of vertex disjoint paths occurring as induced subgraphs of that cover all the vertices of . The path on vertices. The complement of (universe may be set of edges in a graph or index set , the rank-spread of simple graph at vertex . The set of symmetric matrices of simple graph , i.e.,     if and only if Note may have loops and the diagonal zero-nonzero pattern is restricted by .     is a symmetric     matrix over     and     is positive semidefinite ( is simple.)     is a symmetric matrix over F    , if    , then     and ( is simple and no requirement that the entry corresponding to an edge be nonzero.) The set of real symmetric matrices. The spectrum of , i.e., the multiset of the roots of the characteristic polynomial in the algebraic closure of , where . Clique covering number of . The smallest number of cliques in a clique covering of  . The matrix of indeterminates for a loop-tree , i.e., for , let be independent indeterminates and and , and let the entries that do not correspond to edges be 0. The parameter of simple graph is the maximum multiplicity of 0 as an eigenvalue among matrices that satisfy: . satisfies the Strong Arnold Hypothesis.

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