For a graph G of order n, the maximum nullity of G is defined to be the largest possible nullity over all real symmetric n × n matrices A whose (i, j)th entry (for i 6= j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. Maximum nullity and the related parameter minimum rank of the same set of matrices have been studied extensively. A new parameter, generic maximum nullity, is ...

For a graph G of order n, the maximum nullity of G is defined to be the largest possible nullity over all real symmetric n × n matrices A whose (i, j)th entry (for i 6= j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. Maximum nullity and the related parameter minimum rank of the same set of matrices have been studied extensively. A new parameter, maximum generic nullity, is ...

The New Romanian Civil code (Law no. 287 of 2009), that has entered into force on October 1, 2011, is an innovative, reforming code, both in respect of civil legal relations, as well as of legal relations of family law and of commercial law, aspects that are incorporated in its structure. With regard to the legal relations of family law, that naturally belong to the Civil code, the current regu...

The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. In this paper we show the expression of the nullity and nullity set of unicyclic graphs with n vertices and girth r, and characterize the unicyclic graphs with extremal nullity.

To any graph G we can associate a simplicial complex ∆(G) whose simplices are the complete subgraphs of G, and thus we say that G is contractible whenever ∆(G) is so. We study the relationship between contractibility and K-nullity of G, where G is called K-null if some iterated clique graph of G is trivial. We show that there are contractible graphs which are not K-null, and that any graph whos...

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 bicy...

Bounds for the ranks of upper-right submatrices of a generalized inverse of a strictly lower k-banded matrix are obtained. It is shown that such ranks can be exactly predicted under some conditions. The proof uses the Nullity Theorem and bordering technique for generalized inverse.

Let A be the adjacency matrix of a graph Γ . The nullity of A (that is, the dimension of the nullspace of A), when viewed as a matrix over a field of prime characteristic p, is called the p-nullity of Γ . We present several families of arc-transitive graphs with arbitrarily large p-nullity. We also show that the p-nullity of a vertextransitive graph of order a power of p is zero, provided that ...

Let G be a graph with n vertices. The nullity of G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. In this paper, we characterize the graphs (resp. bipartite graphs) with pendent vertices and nullity η, where 0 < η ≤ n. Moreover, the minimum (resp. maximum) number of edges for all (connected) graphs with pendent vertices and nullity η are determined, and the extrem...