The rank+nullity theorem states that, if T is a linear transformation from a finite-dimensional vector space V to a finite-dimensional vector space W , then dim(V ) = rank(T ) + nullity(T ), where rank(T ) = dim(im(T )) and nullity(T ) = dim(ker(T )). The proof treated here is standard; see, for example, [14]: take a basis A of ker(T ) and extend it to a basis B of V , and then show that dim(im...

The object of this offering article is to introduce the notion *- Miao-Tam critical equation on almost contact metric manifolds and it studied Kenmotsu with some nullity condition. It proved that if a (2n + 1)-dimensional (k, µ)!-almost manifold (M, g) satisfies *-Miao-Tam equation, then *-Ricci flat locally isometric product space. Finally, result verified by an example.

We generalize the Tutte polynomial of a matroid to morphism matroids via K-theory flag varieties. introduce two different generalizations, and demonstrate that each has its own merits, where trade-off is between ease combinatorics geometry. One generalization recovers Las Vergnas matroids, which admits corank-nullity formula deletion-contraction recursion. The other does not, but better reflect...

A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. We prove that for a given nullity more than 1, there are only finitely many integral trees. It is also shown that integral trees with nullity 2 and 3 are unique.

Let G be a graph with n(G) vertices and m(G) be its matching number. The nullity of G, denoted by η(G), is the multiplicity of the eigenvalue zero of adjacency matrix of G. It is well known that if G is a tree, then η(G) = n(G)− 2m(G). Guo et al. [Jiming GUO, Weigen YAN, Yeongnan YEH. On the nullity and the matching number of unicyclic graphs. Linear Alg. Appl., 2009, 431: 1293–1301] proved tha...

For any matroid M realizable over Q, we give a combinatorial interpretation of the Tutte polynomial T M (x; y) which generalizes many of its known interpretations and specializations, including Tutte's coloring and ow interpretations of T M (1 ? t; 0); T M (0; 1 ? t), Crapo and Rota's nite eld interpretation of T M (1 ? q k ; 0), the interpretation in terms of the Whitney corank-nullity polynom...