Sharp spectral inequalities for connected bipartite graphs with maximal Q-index

q - regular bipartite graphs

An explicit construction of a family of binary LDPC codes called LU(3, q), where q is a power of a prime, was recently given. A conjecture was made for the dimensions of these codes when q is odd. The conjecture is proved in this note. The proof involves the geometry of a 4-dimensional symplectic vector space and the action of the symplectic group and its subgroups.

Sharp Inequalities for Maximal Functions Associated with General Measures

Sharp weak type (1, 1) and L p estimates in dimension one are obtained for uncentered maximal functions associated with Borel measures which do not necessarily satisfy a doubling condition. In higher dimensions uncentered maximal functions fail to satisfy such estimates. Analogous results for centered maximal functions are given in all dimensions.

The Sharp Lower Bound for the Spectral Radius of Connected Graphs with the Independence Number

In this paper, we investigate some properties of the Perron vector of connected graphs. These results are used to characterize all extremal connected graphs which attain the minimum value among the spectral radii of all connected graphs with order n = kα and the independence number α. Moreover, all extremal graphs which attain the maximum value among the spectral radii of clique trees with orde...

The maximal total irregularity of some connected graphs

The total irregularity of a graph G is defined as 〖irr〗_t (G)=1/2 ∑_(u,v∈V(G))▒〖|d_u-d_v |〗, where d_u denotes the degree of a vertex u∈V(G). In this paper by using the Gini index, we obtain the ordering of the total irregularity index for some classes of connected graphs, with the same number of vertices.

Weighted Norm Inequalities for the Local Sharp Maximal Function