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Subgraph densities have been defined, and served as basic tools, both in the case of graphons (limits of dense graph sequences) and graphings (limits of bounded-degree graph sequences). While limit objects have been described for the middle ranges, the notion of subgraph densities in these limit objects remains elusive. We define subgraph densities in the orthogonality graphs on the unit spheres in dimension $d$, under appropriate sparsity condition on the subgraphs. These orthogonality graphs exhibit the main difficulties of defining subgraphs the middle range, and so we expect their study to serve as a key example to defining subgraph densities in more general Markov spaces. The problem can also be formulated as defining and computing random orthogonal representations of graphs. Orthogonal representations have played a role in information theory, optimization, rigidity theory and quantum physics, so to study random ones may be of interest from the point of view of these applications as well.
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We present a detailed probabilistic and structural analysis of the set of weighted homomorphisms from the discrete torus $mathbb{Z}_m^n$, where $m$ is even, to any fixed graph: we show that the corresponding probability distribution on such homomorphisms is close to a distribution defined constructively as a certain random perturbation of some dominant phase. This has several consequences, including solutions (in a strong form) to conjectures of Engbers and Galvin and a conjecture of Kahn and Park. Special cases include sharp asymptotics for the number of independent sets and the number of proper $q$-colourings of $mathbb{Z}_m^n$ (so in particular, the discrete hypercube). We give further applications to the study of height functions and (generalised) rank functions on the discrete hypercube and disprove a conjecture of Kahn and Lawrenz. For the proof we combine methods from statistical physics, entropy and graph containers and exploit isoperimetric and algebraic properties of the torus.
Lattice-based cryptography is not only for thwarting future quantum computers, and is also the basis of Fully Homomorphic Encryption. Motivated from the advantage of graph homomorphisms we combine graph homomorphisms with graph total colorings together for designing new types of graph homomorphisms: totally-colored graph homomorphisms, graphic-lattice homomorphisms from sets to sets, every-zero graphic group homomorphisms from sets to sets. Our graph-homomorphism lattices are made up by graph homomorphisms. These new homomorphisms induce some problems of graph theory, for example, Number String Decomposition and Graph Homomorphism Problem.
The random reversal graph offers new perspectives, allowing to study the connectivity of genomes as well as their most likely distance as a function of the reversal rate. Our main result shows that the structure of the random reversal graph changes dramatically at $lambda_n=1/binom{n+1}{2}$. For $lambda_n=(1-epsilon)/binom{n+1}{2}$, the random graph consists of components of size at most $O(nln(n))$ a.s. and for $(1+epsilon)/binom{n+1}{2}$, there emerges a unique largest component of size $sim wp(epsilon) cdot 2^ncdot n$!$ a.s.. This giant component is furthermore dense in the reversal graph.
A signed graph $Gamma(G)$ is a graph with a sign attached to each of its edges, where $G$ is the underlying graph of $Gamma(G)$. The energy of a signed graph $Gamma(G)$ is the sum of the absolute values of the eigenvalues of the adjacency matrix $A(Gamma(G))$ of $Gamma(G)$. The random signed graph model $mathcal{G}_n(p, q)$ is defined as follows: Let $p, q ge 0$ be fixed, $0 le p+q le 1$. Given a set of $n$ vertices, between each pair of distinct vertices there is either a positive edge with probability $p$ or a negative edge with probability $q$, or else there is no edge with probability $1-(p+ q)$. The edges between different pairs of vertices are chosen independently. In this paper, we obtain an exact estimate of energy for almost all signed graphs. Furthermore, we establish lower and upper bounds to the energy of random multipartite signed graphs.
A defensive $k$-alliance in a graph is a set $S$ of vertices with the property that every vertex in $S$ has at least $k$ more neighbors in $S$ than it has outside of $S$. A defensive $k$-alliance $S$ is called global if it forms a dominating set. In this paper we study the problem of partitioning the vertex set of a graph into (global) defensive $k$-alliances. The (global) defensive $k$-alliance partition number of a graph $Gamma=(V,E)$, ($psi_{k}^{gd}(Gamma)$) $psi_k^{d}(Gamma)$, is defined to be the maximum number of sets in a partition of $V$ such that each set is a (global) defensive $k$-alliance. We obtain tight bounds on $psi_k^{d}(Gamma)$ and $psi_{k}^{gd}(Gamma)$ in terms of several parameters of the graph including the order, size, maximum and minimum degree, the algebraic connectivity and the isoperimetric number. Moreover, we study the close relationships that exist among partitions of $Gamma_1times Gamma_2$ into (global) defensive $(k_1+k_2)$-alliances and partitions of $Gamma_i$ into (global) defensive $k_i$-alliances, $iin {1,2}$.
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