Maximizing the degree powers of graphs with fixed size

We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince a referee that they have a colouring of the graph.After discussing this notion from first principles, we go on to establish relations with the clique number and orthogonal representations of the graph. We also prove several general facts about this graph parameter and find large separations between the clique number and the quantum chromatic number by looking at random graphs. Finally, we show that there can be no separation between classical and quantum chromatic number if the latter is $2$, nor if it is $3$ in a restricted quantum model; on the other hand, we exhibit a graph on $18$ vertices and $44$ edges with chromatic number $5$ and quantum chromatic number $4$.

Graph parameters such as the clique number and the chromatic number are central in many areas, ranging from computer networks to linguistics to computational neuroscience to social networks. In particular, the chromatic number of a graph can be applied in solving practical tasks as diverse as pattern matching, scheduling jobs to machines, allocating registers in compiler optimization, and even solving Sudoku puzzles. Typically, however, the underlying graphs are subject to (often minor) changes. To make these applications of graph parameters robust, it is important to know which graphs are stable in the sense that adding or deleting single edges or vertices does not change them. We initiate the study of stability of graphs in terms of their computational complexity. We show for various central graph parameters that deciding the stability of a given graph is complete for Θ2p, a well-known complexity class in the second level of the polynomial hierarchy.

It is well-known that eigenvalues of graphs can be used to describe structural properties and parameters of graphs. A theorem of Nosal and Nikiforov states that if $G$ is a triangle-free graph with $m$ edges, then $\lambda (G)\le \sqrt{m}$, equality holds if and only if $G$ is a complete bipartite graph. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] proved a generalization for non-bipartite triangle-free graphs. Moreover, Zhai and Shu [Discrete Math. 345 (2022)] presented a further improvement. In this paper, we present an alternative method for proving the improvement by Zhai and Shu. Furthermore, the method can allow us to give a refinement on the result of Zhai and Shu for non-bipartite graphs without short odd cycles.

Spectral radius of strongly connected digraphs

Let us consider the finite monogenic semigroup S M with zero having elements { x , x 2 , x 3 , … , x n } . There exists an undirected graph Γ ( S M ) associated with S M whose vertices are the non-zero elements x , x 2 , x 3 , … , x n and, f or 1 ≤ i , j ≤ n , any two distinct vertices x i and x j are adjacent if i + j > n . In this paper, the diameter, girth, maximum and minimum degrees, domination number, chromatic number, clique number, degree sequence, irregularity index and also perfectness of Γ ( S M ) have been established. In fact, some of the results obtained in this section are sharper and stricter than the results presented in DeMeyer et al. (Semigroup Forum 65:206-214, 2002). Moreover, the number of triangles for this special graph has been calculated. In the final part of the paper, by considering two (not necessarily different) graphs Γ ( S M 1 ) and Γ ( S M 2 ) , we present the spectral properties to the Cartesian product Γ ( S M 1 ) □ Γ ( S M 2 ) . MSC:05C10, 05C12, 06A07, 15A18, 15A36.

For a hypergraph H on the vertex set {1,...,n}, a distribution D = (D_1,...,D_n) over {0,1}^n is H-wise independent if every restriction of D to indices which form an edge in H is uniform. This generalizes the notion of k-wise independence obtained by taking H to be the complete n vertex k-uniform hypergraph. This generalization was studied by Schulman (STOC 1992), who presented constructions of H-wise independent distributions that are linear, i.e., the samples are strings of inner products (over F2) of a fixed set of vectors with a uniformly chosen random vector. Let l(H) denote the minimum possible size of a sample space of a uniform H-wise independent distribution. The l parameter is well understood for the special case of k-wise independence. In this work we study the notion of H-wise independence and the l parameter for general graphs and hypergraphs. For graphs, we show how the l parameter relates to standard graph parameters (e.g., clique number, chromatic number, Lovasz theta function, minrank). We derive algorithmic and hardness results for this parameter as well as an explicit construction of graphs G for which l(G) is exponentially smaller than the size of the sample space of any linear G-wise independent distribution. For hypergraphs, we study the problem of testing whether a given distribution is H-wise independent, generalizing results of Alon et al. (STOC 2007).

We say a graph has property $\mathcal{P}_{g,p}$ when it is an induced subgraph of the curve graph of a surface of genus $g$ with $p$ punctures. Two well-known graph invariants, the chromatic and clique numbers, can provide obstructions to $\mathcal{P}_{g,p}$. We introduce a new invariant of a graph, the \emph{nested complexity length}, which provides a novel obstruction to $\mathcal{P}_{g,p}$. For the curve graph this invariant captures the topological complexity of the surface in graph-theoretic terms; indeed we show that its value is $6g-6+2p$, i.e. twice the size of a maximal multicurve on the surface. As a consequence we show that large `half-graphs' do not have $\mathcal{P}_{g,p}$, and we deduce quantitatively that almost all finite graphs which pass the chromatic and clique tests do not have $\mathcal{P}_{g,p}$. We also reinterpret our obstruction in terms of the first-order theory of the curve graph, and in terms of RAAG subgroups of the mapping class group (following Kim and Koberda). Finally, we show that large complete multipartite graphs cannot have $\mathcal{P}_{g,p}$. This allows us to compute the upper density of the curve graph, and to conclude that clique size, chromatic number, and nested complexity length are not sufficient to determine $\mathcal{P}_{g,p}$.

The multiplicative sum Zagreb index is a modified version of the well-known Zagreb indices. The multiplicative sum Zagreb index of a graph G is the product of the sums of the degrees of pairs of adjacent vertices. The mathematical properties of the multiplicative sum Zagreb index of graphs with given graph parameters deserve further study, as they can be used to detect chemical compounds and study network structures in mathematical chemistry. Therefore, in this paper, the maximal and minimal values of the multiplicative sum Zagreb indices of graphs with a given clique number are presented. Furthermore, the corresponding extremal graphs are characterized.

Circular chromatic numbers of Mycielski's graphs

Nordhaus–Gaddum inequalities for the fractional and circular chromatic numbers

For a nontrivial connected graph $G$ of order $n$, the detour distance $D(u,v)$ between two vertices $u$ and $v$ in $G$ is the length of a longest $u-v$ path in $G$. Detour distance is a metric on the vertex set of $G$. For each integer $k$ with $1\le k\le n-1$, a coloring $c\colon V(G)\to \mathbb N$ is a $k$-metric coloring of $G$ if $|c(u)-c(v)|+D(u,v)\ge k+1$ for every two distinct vertices $u$ and $v$ of $G$. The value $\chi _m^k(c)$ of a $k$-metric coloring $c$ is the maximum color assigned by $c$ to a vertex of $G$ and the $k$-metric chromatic number $\chi _m^k(G)$ of $G$ is the minimum value of a $k$-metric coloring of $G$. For every nontrivial connected graph $G$ of order $n$, $\chi _m^1(G)\le \chi _m^2(G)\le \cdots \le \chi _m^{n-1}(G)$. Metric chromatic numbers provide a generalization of several well-studied coloring parameters in graphs. Upper and lower bounds have been established for $\chi _m^k(G)$ in terms of other graphical parameters of a graph $G$ and exact values of $k$-metric chromatic numbers have been determined for complete multipartite graphs and cycles. For a nontrivial connected graph $G$, the anti-diameter ${\rm adiam}(G)$ is the minimum detour distance between two vertices of $G$. We show that the ${\rm adiam}(G)$-metric chromatic number of a graph $G$ provides information on the Hamiltonian properties of the graph and investigate realization results and problems on this parameter.

Homomorphisms of binary Cayley graphs

Extremal graph problems with symmetrical extremal graphs. Additional chromatic conditions

A graph is called $k$ -critical if its chromatic number is $k$ but every proper subgraph has chromatic number less than $k$ . An old and important problem in graph theory asks to determine the maximum number of edges in an $n$ -vertex $k$ -critical graph. This is widely open for every integer $k\geq 4$ . Using a structural characterisation of Greenwell and Lovász and an extremal result of Simonovits, Stiebitz proved in 1987 that for $k\geq 4$ and sufficiently large $n$ , this maximum number is less than the number of edges in the $n$ -vertex balanced complete $(k-2)$ -partite graph. In this paper, we obtain the first improvement in the above result in the past 35 years. Our proofs combine arguments from extremal graph theory as well as some structural analysis. A key lemma we use indicates a partial structure in dense $k$ -critical graphs, which may be of independent interest.

Graph theory can give a representation of abstract mathematical systems such as groups or rings. We have many graph representations for a group, in this study we use the coprime graph representation for a generalized quaternion group to find the numerical invariants of the graph, which are the clique number and the chromatic number. The main results obtained from this study are the clique number of the coprime graph representation for the generalized quaternion group is equal to the chromatic number of the coprime graph representation for the generalized quaternion group for each case of the order.