Related papers: Computing the Grothendieck constant of some graph …

The quadratic embedding constant (QEC) of a graph $G$ is a new numeric invariant, which is defined in terms of the distance matrix and is denoted by $\mathrm{QEC}(G)$. By observing graph structure of the maximal cliques (clique graph), we…

The Grothendieck constant $K_{G}$ is a fundamental quantity in functional analysis, with important connections to quantum information, combinatorial optimization, and the geometry of Banach spaces. Despite decades of study, the value of…

Semidefinite relaxations are a powerful tool for approximately solving combinatorial optimization problems such as MAX-CUT and the Grothendieck problem. By exploiting a bounded rank property of extreme points in the semidefinite cone, we…

We offer a new, gradual approach to the largest girth problem for cubic graphs. It is easily observed that the largest possible girth of all $n$-vertex cubic graphs is attained by a $2$-connected graph $G=(V,E)$. By Petersen's graph…

The QE constant of a finite connected graph $G$, denoted by $\mathrm{QEC}(G)$, is by definition the maximum of the quadratic function associated to the distance matrix on a certain sphere of codimension two. We prove that the QE constants…

The quadratic embedding constant (QEC) of a finite, simple, connected graph $G$ is the maximum of the quadratic form of the distance matrix of $G$ on the subset of the unit sphere orthogonal to the all-ones vector. The study of these QECs…

Motivated by understanding the quality of tractable convex relaxations of intractable polytopes, Ko et al. gave a closed-form expression for the volume of a standard relaxation $\mathscr{Q}(G)$ of the boolean quadric polytope (also known as…

Let ${\rm ind}(G)$ be the number of independent sets in a graph $G$. We show that if $G$ has maximum degree at most $5$ then $$ {\rm ind}(G) \leq 2^{{\rm iso}(G)} \prod_{uv \in E(G)} {\rm ind}(K_{d(u),d(v)})^{\frac{1}{d(u)d(v)}} $$ (where…

Let $G = (V,E)$ be a graph and $k \ge 0$ an integer. A $k$-independent set $S \subseteq V$ is a set of vertices such that the maximum degree in the graph induced by $S$ is at most $k$. With $\alpha_k(G)$ we denote the maximum cardinality of…

As part of the search for the value of the smallest upper bound of the best constant for the famous Grothendieck inequality, the so-called Grothendieck constant (a hard open problem - unsolved since 1953), we provide a further approach,…

Given an integer $k$, deciding whether a graph has a clique of size $k$ is an NP-complete problem. Wilf's inequality provides a spectral bound for the clique number of simple graphs. Wilf's inequality is stated as follows: $\frac{n}{n -…

For a graph $G=(V,E)$ with $v(G)$ vertices the partition function of the random cluster model is defined by $$Z_G(q,w)=\sum_{A\subseteq E(G)}q^{k(A)}w^{|A|},$$ where $k(A)$ denotes the number of connected components of the graph $(V,A)$.…

Within the framework of the search for the still unknown exact value of the real and complex Grothendieck constant $K_G^\mathbb{F}$ in the famous Grothendieck inequality (unsolved since 1953), where $\mathbb{F}$ denotes either the real or…

We study a new geometric graph parameter $\egd(G)$, defined as the smallest integer $r\ge 1$ for which any partial symmetric matrix which is completable to a correlation matrix and whose entries are specified at the positions of the edges…

For every graph $G$, let $\omega(G)$ be the largest size of complete subgraph in $G$. This paper presents a simple algorithm which, on input a graph $G$, a positive integer $k$ and a small constant $\epsilon>0$, outputs a graph $G'$ and an…

Let $G$ be a graph with edge set $E(G)$. Denote by $d_w$ the degree of a vertex $w$ of $G$. The sigma index of $G$ is defined as $\sum_{uv\in E(G)}(d_u-d_v)^2$. A connected graph of order $n$ and size $n+k-1$ is known as a connected…

We study sufficient conditions for the generic rigidity of a graph $G$ expressed in terms of (i) its minimum degree $\delta(G)$, or (ii) the parameter $\eta(G)=\min_{uv\notin E}(\deg(u)+\deg(v))$. For each case, we seek the smallest…

The K_4-free process starts with the empty graph on n vertices and at each step adds a new edge chosen uniformly at random from all remaining edges that do not complete a copy of K_4. Let G be the random maximal K_4-free graph obtained at…

For a graph $G$, a vertex subset $S$ is called a maximum generalized $k$-independent set if the induced subgraph $G[S]$ does not contain a $k$-tree as its subgraph, and the subset has maximum cardinality. The generalized $k$-independence…

Let $G$ be a connected graph of order $n$.The Wiener index $W(G)$ of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. In this paper we show that the well-known upper bound $\big( \frac{n}{\delta+1}+2\big) {n…