Related papers: Numerical radius and zero pattern of matrices
We determine the maximum possible number of edges of a graph with $n$ vertices, matching number at most $s$ and clique number at most $k$ for all admissible values of the parameters.
In this work, we give the sharp upper bound for the number of cliques in graphs with bounded odd circumferences. This generalized Tur\'an-type result is an extension of the celebrated Erd\H{o}s and Gallai theorem and a strengthening of…
We present new bounds for the numerical radius of bounded linear operators and $2\times 2$ operator matrices. We apply upper bounds for the numerical radius to the Frobenius companion matrix of a complex monic polynomial to obtain new…
In this paper, we derive nearly tight probabilistic norm bounds for a class of random matrices we call graph matrices. While the classical case of symmetric matrices with independent random entries (Wigner's matrices) is a special case, in…
The fundamental theorem of Tur\'{a}n from Extremal Graph Theory determines the exact bound on the number of edges $t_r(n)$ in an $n$-vertex graph that does not contain a clique of size $r+1$. We establish an interesting link between…
In this paper, we give upper and lower bounds for the spectral radius of a nonnegative irreducible matrix and characterize the equality cases. These bounds theoretically improve and generalize some known results of Duan et al.[X. Duan, B.…
Let $\cal H$ be a family of graphs. The Tur\'an number ${\rm ex}(n,{\cal H})$ is the maximum possible number of edges in an $n$-vertex graph which does not contain any member of $\cal H$ as a subgraph. As a common generalization of…
Several spectral radii formulas for infinite bounded nonnegative matrices in max algebra are obtained. We also prove some Perron-Frobenius type results for such matrices. In particular, we obtain results on block triangular forms, which are…
We determine new upper bounds for the clique numbers of strongly regular graphs in terms of their parameters. These bounds improve on the Delsarte bound for infinitely many feasible parameter tuples for strongly regular graphs, including…
Generalized Tur\'an problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem is maximizing the number of cliques of size $t$ in a graph of a fixed order that does not contain…
A zero-one matrix $M$ is said to contain another zero-one matrix $A$ if we can delete some rows and columns of $M$ and replace some $1$-entries with $0$-entries such that the resulting matrix is $A$. The extremal number of $A$, denoted…
We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$. For $(r-1)|M|\ge n$, the $(r-1)$-partite Turan graph turns out to be the unique…
The numerical radius of a matrix is a scalar quantity that has many applications in the study of matrix analysis. Due to the difficulty in computing the numerical radius, inequalities bounding it have received a considerable attention in…
An abstract simplicial complex $\mathbf{F}$ is a non-uniform hypergraph without isolated vertices, whose edge set is closed under taking subsets. The extremal number $\mathrm{ex}(n,\mathbf{F})$ is the maximum number of edges in an…
For a square matrix, the range of its Rayleigh quotients is known as the numerical range, which is a compact and convex set by the Toeplitz-Hausdorff theorem. The largest value and the smallest boundary value (in magnitude) of this convex…
Consider a graph on the non-singular matrices over a finite field, in which two distinct non-singular matrices are joined by an edge whenever their sum is singular. We prove an upper bound for the independence number of this graph. As a…
We give upper and lower bounds for the spectral radius of a nonnegative matrix by using its average 2-row sums, and characterize the equality cases if the matrix is irreducible. We also apply these bounds to various nonnegative matrices…
We use two variational techniques to prove upper bounds for sums of the lowest several eigenvalues of matrices associated with finite, simple, combinatorial graphs. These include estimates for the adjacency matrix of a graph and for both…
The aim of this work is to investigate the nonnegative signed domination number $\gamma^{NN}_s$ with emphasis on regular, ($r+1$)-clique-free graphs and trees. We give lower and upper bounds on $\gamma^{NN}_s$ for regular graphs and prove…
Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs. The random Tur\'an number $\mathrm{ex}(G^r_{n,p},\mathcal{F})$ is the maximum number of edges in an $\mathcal{F}$-free subgraph of $G^r_{n,p}$, where $G^r_{n,p}$ is the…