In this paper we disprove a conjecture of Lidický and Murphy about the number copies given graph in $K_r$-free give an alternative general conjecture. We also prove asymptotically tight bound on any bipartite radius at most~$2$ triangle-free graph.

Abstract Let $\varphi _1,\ldots ,\varphi _r\in {\mathbb Z}[z_1,\ldots z_k]$ be integral linear combinations of elementary symmetric polynomials with $\text {deg}(\varphi _j)=k_j\ (1\le j\le r)$ , where $1\le k_1<k_2<\cdots <k_r=k$ . Subject to the condition $k_1+\cdots +k_r\ge \tfrac {1}{2}k(k-~1)+2$ we show that there is a paucity nondiagonal solutions Diophantine system _j({\mathbf x...

The feasible region $\Omega_{{\rm ind}}(F)$ of a graph $F$ is the collection points $(x,y)$ in unit square such that there exists sequence graphs whose edge densities approach $x$ and induced $F$-densities $y$. A complete description not known for any with at least four vertices clique or an independent set. provides lot combinatorial information about $F$. For example, supremum $y$ over all $(...

For a graph $G$, let $f(G)$ denote the size of maximum cut in $G$. The problem estimating as function number vertices and edges $G$ has long history was extensively studied last fifty years. In this paper we propose an approach, based on semidefinite programming, to prove lower bounds $f(G)$. We use approach find large cuts graphs with few triangles $K_r$-free graphs.

the inflation $g_{i}$ of a graph $g$ with $n(g)$ vertices and $m(g)$ edges is obtained from $g$ by replacing every vertex of degree $d$ of $g$ by a clique, which is isomorph to the complete graph $k_{d}$, and each edge $(x_{i},x_{j})$ of $g$ is replaced by an edge $(u,v)$ in such a way that $uin x_{i}$, $vin x_{j}$, and two different edges of $g$ are replaced by non-adjacent edges of $g_{i}$. t...

Nearby radio galaxies that contain jets are extensively studied with VLBI, addressing jet launching and the physical mechanisms at play around massive black holes. 3C 84 is unique in this regard, because combination of its proximity large SMBH mass provides a high spatial resolution to resolve complex structure base. For an angular scale 50 ${\mu}$as corresponds 200 - 250 Schwarzschild radii ($...