Stability for large forbidden subgraphs

We extend the classical stability theorem of Erdos and Simonovits for forbidden graphs of logarithmic order.Comment: Some polishing. Updated reference

Raising and lowering operators and their factorization for generalized orthogonal polynomials of hypergeometric type on homogeneous and non-homogeneous lattice

We complete the construction of raising and lowering operators, given in a previous work, for the orthogonal polynomials of hypergeometric type on non-homogeneous lattice, and extend these operators to the generalized orthogonal polynomials, namely, those difference of orthogonal polynomials that satisfy a similar difference equation of hypergeometric type.Comment: LaTeX, 19 pages, (late submission to arXiv.org

Revisiting two classical results on graph spectra

Let mu(G) and mu_min(G) be the largest and smallest eigenvalues of the adjacency matricx of a graph G. We refine quantitatively the following two results on graph spectra. (i) if H is a proper subgraph of a connected graph G, then mu(G)>mu(H). (ii) if G is a connected nonbipartite graph, then mu(G)>-mu_min(G)

Graphs with many copies of a given subgraph

We show that if a graph G of order n contains many copies of a given subgraph H, then it contains a blow-up of H of order log n

Max k-cut and the smallest eigenvalue

Let $G$ be a graph of order $n$ and size $m$, and let $\mathrm{mc}_{k}\left( G\right)$ be the maximum size of a $k$-cut of $G.$ It is shown that $$\mathrm{mc}_{k}\left( G\right) \leq\frac{k-1}{k}\left( m-\frac{\mu_{\min }\left( G\right) n}{2}\right) ,$$ where $\mu_{\min}\left( G\right)$ is the smallest eigenvalue of the adjacency matrix of $G.$ An infinite class of graphs forcing equality in this bound is constructed.Comment: 5 pages. Some typos corrected in v

Beyond graph energy: norms of graphs and matrices

In 1978 Gutman introduced the energy of a graph as the sum of the absolute values of graph eigenvalues, and ever since then graph energy has been intensively studied. Since graph energy is the trace norm of the adjacency matrix, matrix norms provide a natural background for its study. Thus, this paper surveys research on matrix norms that aims to expand and advance the study of graph energy. The focus is exclusively on the Ky Fan and the Schatten norms, both generalizing and enriching the trace norm. As it turns out, the study of extremal properties of these norms leads to numerous analytic problems with deep roots in combinatorics. The survey brings to the fore the exceptional role of Hadamard matrices, conference matrices, and conference graphs in matrix norms. In addition, a vast new matrix class is studied, a relaxation of symmetric Hadamard matrices. The survey presents solutions to just a fraction of a larger body of similar problems bonding analysis to combinatorics. Thus, open problems and questions are raised to outline topics for further investigation.Comment: 54 pages. V2 fixes many typos, and gives some new materia

The trace norm of r-partite graphs and matrices

The trace norm $\left\Vert G\right\Vert _{\ast}$ of a graph $G$ is the sum of its singular values, i.e., the absolute values of its eigenvalues. The norm $\left\Vert G\right\Vert _{\ast}$ has been intensively studied under the name of graph energy, a concept introduced by Gutman in 1978. This note studies the maximum trace norm of $r$-partite graphs, which raises some unusual problems for $r>2$. It is shown that, if $G$ is an $r$-partite graph of order $n,$ then $$\left\Vert G\right\Vert _{\ast}<\frac{n^{3/2}}{2}\sqrt{1-1/r}+\left( 1-1/r\right) n.$$ For some special $r$ this bound is tight: e.g., if $r$ is the order of a symmetric conference matrix, then, for infinitely many $n,$ there is a graph $G\$of order $n$ with $$\left\Vert G\right\Vert _{\ast}>\frac{n^{3/2}}{2}\sqrt{1-1/r}-\left( 1-1/r\right) n.$$Comment: 12 page