[en] For deterministically growing networks, it is a theoretical challenge to determine the topological properties and dynamical processes. In this paper, we study random walks on generalized Koch networks with features that include an initial state that is a globally connected network to r nodes. In each step, every existing node produces m complete graphs. We then obtain the analytical expressions for first passage time (FPT), average return time (ART), i.e. the average of FPTs for random walks from node i to return to the starting point i for the first time, and average sending time (AST), defined as the average of FPTs from a hub node to all other nodes, excluding the hub itself with regard to network parameters m and r. For this family of Koch networks, the ART of the new emerging nodes is identical and increases with the parameters m or r. In addition, the AST of our networks grows with network size N as N ln N and also increases with parameter m. The results obtained in this paper are the generalizations of random walks for the original Koch network. (paper)

[en] A graph is said to be total-colored if all the edges and the vertices of the graph are colored. A total-colored graph is total-rainbow connected if any two vertices of the graph are connected by a path whose edges and internal vertices have distinct colors. For a connected graph G, the total-rainbow connection number of G, denoted by trc(G), is the minimum number of colors required in a total-coloring of G to make G total-rainbow connected. In this paper, we first characterize the graphs having large total-rainbow connection numbers. Based on this, we obtain a Nordhaus–Gaddum-type upper bound for the total-rainbow connection number. We prove that if G and $\stackrel{¯<!--\; ??\; -->}{G}$ are connected complementary graphs on n vertices, then $trc(G)+trc(\stackrel{¯<!--\; ??\; -->}{G})\le <!--\; ???\; -->2n$ when $n\ge <!--\; ???\; -->6$ and $trc(G)+trc(\stackrel{¯<!--\; ??\; -->}{G})\le <!--\; ???\; -->2n+1$ when $n=5$. Examples are given to show that the upper bounds are sharp for $n\ge <!--\; ???\; -->5$. This completely solves a conjecture in Ma (Res Math 70(1–2):173–182, 2016).

[en] A SYK–like model close to the colored tensor models has recently been proposed [1]. Building on results obtained in tensor models [2], we discuss the complete $1/N$ expansion of the model. We detail the two and four point functions at leading order. The leading order two point function is a sum over melonic graphs, and the leading order relevant four point functions are sums over dressed ladder diagrams. We then show that any order in the $1/N$ series of the two point function can be written solely in term of the leading order two and four point functions. The full $1/N$ expansion of arbitrary correlations can be obtained by similar methods.

[en] Exponential random graph models have attracted significant research attention over the past decades. These models are maximum-entropy ensembles subject to the constraints that the expected values of a set of graph observables are equal to given values. Here we extend these maximum-entropy ensembles to random simplicial complexes, which are more adequate and versatile constructions to model complex systems in many applications. We show that many random simplicial complex models considered in the literature can be casted as maximum-entropy ensembles under certain constraints. We introduce and analyze the most general random simplicial complex ensemble $\mathbf{\Delta <!--\; ??\; -->}$ with statistically independent simplices. Our analysis is simplified by the observation that any distribution $\mathbb{P}(O)$ on any collection of objects $\mathcal{O}=\{O\},$ including graphs and simplicial complexes, is maximum-entropy subject to the constraint that the expected value of $-<!--\; ???\; -->\mathrm{l}\mathrm{n}\mathbb{P}(O)$ is equal to the entropy of the distribution. With the help of this observation, we prove that ensemble $\mathbf{\Delta <!--\; ??\; -->}$ is maximum-entropy subject to the two types of constraints which fix the expected numbers of simplices and their boundaries. (paper)

[en] We introduce a new method for the enumeration of self-avoiding walks based on the lace expansion. We also introduce an algorithmic improvement, called the two-step method, for self-avoiding walk enumeration problems. We obtain significant extensions of existing series on the cubic and hypercubic lattices in all dimensions d ≥ 3: we enumerate 32-step self-avoiding polygons in d = 3, 26-step self-avoiding polygons in d = 4, 30-step self-avoiding walks in d = 3, and 24-step self-avoiding walks and polygons in all dimensions d ≥ 4. We analyze these series to obtain estimates for the connective constant and various critical exponents and amplitudes in dimensions 3 ≤ d ≤ 8. We also provide major extensions of 1/d expansions for the connective constant and for two critical amplitudes

[en] This paper is the second in a series in which we complete the description of the finite vertex stabilizers for connected graphs with projective suborbits and, as a corollary, of the vertex stabilizers for finite connected graphs in groups of automorphisms that act transitively on 2-arcs. In this part we complete the treatment of the collineation case, under the assumption that the suborbit has projective dimension 3, and of the correlation case

[en] This paper describes a teaching method that allows for the fast and early detection of students' conceptions, misconceptions and their development. The empirical study of two examples where the method was applied is reported. The prerequisites for the efficient use of the method are discussed and results of the pilot study of its effectiveness are briefly presented

[en] The energy of a graph G is defined as the sum of the absolute values of all eigenvalues of G and denoted by E(G). Let G be a graph and rank(G) be the rank of the adjacency matrix of G. In this paper we characterize all the graphs with E(G) = rank(G). Among other results we show that apart from a few families of graphs, E(G) ≥ 2max(χ(G), n - χ(G--bar)), where G-bar and χ(G) are the complement and the chromatic number of G, respectively. Moreover some new lower bounds for E(G) in terms of rank(G) are given. (author)

[en] Let R be a ring with unity and I(R)* be the set of all non-trivial left ideals of R. The intersection graph of ideals of R, denoted by G(R), is a graph with the vertex set I(R)* and two distinct vertices I and J are adjacent if and only if I intersection J ≠ 0. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose intersection graphs of ideals are not connected. Also we determine all rings whose clique number of the intersection graphs of ideals are finite. Among other results, it is shown that for every ring, if the clique number of G(R) is finite, then the chromatic number is finite too and if R is a reduced ring both are equal. (author)

[en] The resistance distance between any two vertices of a connected graph is defined as the effective resistance between them in the electrical network constructed from the graph by replacing each edge with a (unit) resistor. Let G+H, G × H, G[H], G⊗H, G ⊠ H, G∘H and G{H} be the join, product, composition, direct product, strong product, corona and rooted product of two graphs G and H, respectively. In this paper, formulae for resistance distances of these composite graphs are given in terms of parameters of the parent graphs, and some properties are established. Explicit formulae are obtained for resistance distances of some classes of graphs, including rook graphs, diagonal mesh graphs and generalized double graphs. (paper)