Paths and related graphs as star complements for 0 or $$-\,1$$ in regular graphs

The well known “real-life examples” of small world graphs, including the graph of binary relation: “two persons on the earth know each other” contains cliques, so they have cycles of order 3 and 4. Some problems of Computer Science require explicit construction of regular algebraic graphs with small diameter but without small cycles. The well known examples here are generalised polygons, which are small world algebraic graphs i.e. graphs with the diameter d≤clog k−1(v), where v is order, k is the degree and c is the independent constant, semiplanes (regular bipartite graphs without cycles of order 4); graphs that can be homomorphically mapped onto the ordinary polygons. The problem of the existence of regular graphs satisfying these conditions with the degree ≥k and the diameter ≥d for each pair k≥3 and d≥3 is addressed in the paper. This problem is positively solved via the explicit construction. Generalised Schubert cells are defined in the spirit of Gelfand-Macpherson theorem for the Grassmanian. Constructed graph, induced on the generalised largest Schubert cells, is isomorphic to the well-known Wenger’s graph. We prove that the family of edge-transitive q-regular Wenger graphs of order 2qn, where integer n≥2 and q is prime power, q≥n, q>2 is a family of small world semiplanes. We observe the applications of some classes of small world graphs without small cycles to Cryptography and Coding Theory.

Codes for distributed storage from 3-regular graphs

Let G =(V, E) be finite and simple graphs with vertex set V(G) and edge set E(G). A graph G is called super edge-magic if there exists a bijection f: V(G) ∪ E(G) → {1, 2, ⋯, |V(G)| + |E(G)|} and f(V(G)) = {1, 2, ⋯, |V(G)|} such that f(x) + f(xy) + f(y) is a constant for every edge xy ∈ E(G). A graph G with isolated vertices is called pseudo super edge-magic if there exists a bijection f: V(G) → {1, 2, ⋯, |V(G)|} such that the set {f(x) + f(y) ∶ xy ∈ E(G)} ∪ {2f(x) ∶ deg(x) = 0} consist of |E(G)| + |{x ∈ V(G) ∶ deg(x) = 0}| consecutive integers. In this paper, we construct (pseudo) super edge-magic 2-regular graphs from a super edge-magic cycle by using normalized Kotzig arrays. We also show that the graph C3 ∪ Cn ∪ K1 is pseudo super edge-magic for n ≡ 1(mod 4). By this result, we obtain some new classes of super edge-magic 2-regular graphs. In addition, we show that union of cycles and paths are super edge-magic.

Regular graphs are considered, whose automorphism groups are permutation representations P of the orthogonal groups in various dimensions over GF(2). Vertices and adjacencies are defined by quadratic forms, and after graphical displays of the trivial isomorphisms between the symmetric groups S2, S3, S5, S6 and corresponding orthogonal groups, a 28-vertex graph is constructed that displays the isomorphism between S8 and o +6 (2). Explored next are the eigenvalues and constituent idempotent matrices of the (−1,1)-adjacency matrix A of each of the orthogonal graphs, and the commuting ring R of the rank three permutation representation P of its automorphism group. Formulas are obtained for splitting into its irreducible characters χ(i) the permutation character χ of P, by expressing the class sums Bλ of P in terms of the identity matrix and the (0,1)-matrices H and K obtained from the adjacency matrix A=H − K.

Unsupervised spatial representation learning aims to automatically identify effective features of geographic entities (i.e., regions) from unlabeled yet structural geographical data. Existing network embedding methods can partially address the problem by: (1) regarding a region as a node in order to reformulate the problem into node embedding; (2) regarding a region as a graph in order to reformulate the problem into graph embedding. However, these studies can be improved by preserving (1) intra-region geographic structures, which are represented by multiple spatial graphs, leading to a reformulation of collective learning from relational graphs; (2) inter-region spatial autocorrelations, which are represented by pairwise graph regularization, leading to a reformulation of adversarial learning. Moreover, field data in real systems are usually lack of labels, an unsupervised fashion helps practical deployments. Along these lines, we develop an unsupervised Collective Graph-regularized dual-Adversarial Learning (CGAL) framework for multi-view graph representation learning and also a Graph-regularized dual-Adversarial Learning (GAL) framework for single-view graph representation learning. Finally, our experimental results demonstrate the enhanced effectiveness of our method.

In the present article, statistical properties regarding the topology and standard percolation on relative neighborhood graphs (RNGs) for planar sets of points, considering the Euclidean metric, are put under scrutiny. RNGs belong to the family of "proximity graphs"; i.e., their edgeset encodes proximity information regarding the close neighbors for the terminal nodes of a given edge. Therefore they are, e.g., discussed in the context of the construction of backbones for wireless ad hoc networks that guarantee connectedness of all underlying nodes. Here, by means of numerical simulations, we determine the asymptotic degree and diameter of RNGs and we estimate their bond and site percolation thresholds, which were previously conjectured to be nontrivial. We compare the results to regular 2D graphs for which the degree is close to that of the RNG. Finally, we deduce the common percolation critical exponents from the RNG data to verify that the associated universality class is that of standard 2D percolation.

Recently, Araujo et al. [Manuscript in preparation, 2017] introduced the notion of Cycle Convexity of graphs. In their seminal work, they studied the graph convexity parameter called hull number for this new graph convexity they proposed, and they presented some of its applications in Knot theory. Roughly, the tunnel number of a knot embedded in a plane is upper bounded by the hull number of a corresponding planar 4-regular graph in cycle convexity. In this paper, we go further in the study of this new graph convexity and we study the interval number of a graph in cycle convexity. This parameter is, alongside the hull number, one of the most studied parameters in the literature about graph convexities. Precisely, given a graph G, its interval number in cycle convexity, denoted by $in_{cc} (G)$, is the minimum cardinality of a set S ⊆ V (G) such that every vertex w ∈ V (G) \ S has two distinct neighbors u, v ∈ S such that u and v lie in same connected component of G[S], i.e. the subgraph of G induced by the vertices in S.In this work, first we provide bounds on $in_{cc} (G)$ and its relations to other graph convexity parameters, and explore its behavior on grids. Then, we present some hardness results by showing that deciding whether $in_{cc} (G)$ ≤ k is NP-complete, even if G is a split graph or a bounded-degree planar graph, and that the problem is W[2]-hard in bipartite graphs when k is the parameter. As a consequence, we obtainthat $in_{cc} (G)$ cannot be approximated up to a constant factor in the classes of split graphs and bipartite graphs (unless P = N P ).On the positive side, we present polynomial-time algorithms to compute $in_{cc} (G)$ for outerplanar graphs, cobipartite graphs and interval graphs. We also present fixed-parameter tractable (FPT) algorithms to compute it for (q, q − 4)-graphs when q is the parameter and for general graphs G when parameterized either by the treewidth or the neighborhood diversity of G.Some of our hardness results and positive results are not known to hold for related graph convexities and domination problems. We hope that the design of our new reductions and polynomial-time algorithms can be helpful in order to advance in the study of related graph problems.

The paper is devoted to the study of a linguistic dynamical system of dimension n ≥ 2 over an arbitrary commutative ring K, i.e., a family F of nonlinear polynomial maps f α : K n → K n depending on “time” α ∈ {K − 0} such that f α −1 = f −αM, the relation f α1 (x) = f α2 (x) for some x ∈ Kn implies α1 = α2, and each map f α has no invariant points. The neighborhood {f α (υ)∣α ∈ K − {0}} of an element v determines the graph Γ(F) of the dynamical system on the vertex set Kn. We refer to F as a linguistic dynamical system of rank d ≥ 1 if for each string a = (α1, υ, α2), s ≤ d, where αi + αi+1 is a nonzero divisor for i = 1, υ, d − 1, the vertices υ a = f α1 × ⋯ × f αs (υ) in the graph are connected by a unique path. For each commutative ring K and each even integer n ≠= 0 mod 3, there is a family of linguistic dynamical systems Ln(K) of rank d ≥ 1/3n. Let L(n, K) be the graph of the dynamical system Ln(q). If K = Fq, the graphs L(n, Fq) form a new family of graphs of large girth. The projective limit L(K) of L(n, K), n → ∞, is well defined for each commutative ring K; in the case of an integral domain K, the graph L(K) is a forest. If K has zero divisors, then the girth of K drops to 4. We introduce some other families of graphs of large girth related to the dynamical systems Ln(q) in the case of even q. The dynamical systems and related graphs can be used for the development of symmetric or asymmetric cryptographic algorithms. These graphs allow us to establish the best known upper bounds on the minimal order of regular graphs without cycles of length 4n, with odd n ≥ 3. Bibliography: 42 titles.

An edge product cordial labeling is a variant of the well-known cordial labeling. In this paper we characterize graphs admitting an edge product cordial labeling. Using this characterization we investigate the edge product cordiality of broad classes of graphs, namely, dense graphs, dense bipartite graphs, connected regular graphs, unions of some graphs, direct products of some bipartite graphs, joins of some graphs, maximal \(k\)-degenerate and related graphs, product cordial graphs.

While actors in a population can interact with anyone else freely, social relations significantly influence our inclination toward particular individuals. The consequence of such interactions, however, may also form the intensity of our relations established earlier. These dynamical processes are captured via a coevolutionary model staged in multiplex networks with two distinct layers. In a so-called relationship layer, the weights of edges among players may change in time as a consequence of games played in the alternative interaction layer. As an reasonable assumption, bilateral cooperation confirms while mutual defection weakens these weight factors. Importantly, the fitness of a player, which basically determines the success of a strategy imitation, depends not only on the payoff collected from interactions, but also on the individual relationship index calculated from the mentioned weight factors of related edges. Within the framework of weak prisoner's dilemma situation, we explore the potential outcomes of the mentioned coevolutionary process where we assume different topologies for relationship layer. We find that higher average degree of the relationship graph is more beneficial to maintain cooperation in regular graphs, but the randomness of links could be a decisive factor in harsh situations. Surprisingly, a stronger coupling between relationship index and fitness discourage the evolution of cooperation by weakening the direct consequence of a strategy change. To complete our study, we also monitor how the distribution of relationship index vary and detect a strong relation between its polarization and the general cooperation level.

This paper presents an analysis of degree 5 chordal rings, from a network topology point of view. The chordal rings are mainly evaluated with respect to average distance and diameter. We derive approximation expressions for the related ideal graphs, and show that these matches the real chordal rings fairly well. Moreover, the results are compared to that of a reference graph which presents a lower bound for average distance and diameter among all regular graphs of degree 5. It turns out that this reference graph has significantly lower distances than the degree 5 chordal rings. Based on that, we suggest that future research could deal with either finding degree 5 topologies with average distance and diameter closer to these of the reference graph, or to develop more realistic bounds than those presented by these reference graphs.

Defectors’ niches: prisoner's dilemma game on disordered networks

Representing graph families with edge grammars

Let $G$ be a connected graph with a distance matrix $D$. The distance eigenvalues of $G$ are the eigenvalues of $D$, and the distance energy $E_D(G)$ is the sum of its absolute values. The transmission $Tr(v)$ of a vertex $v$ is the sum of the distances from $v$ to all other vertices in $G$. The transmission matrix $Tr(G)$ of $G$ is a diagonal matrix with diagonal entries equal to the transmissions of vertices. The matrices $D^L(G)= Tr(G)-D(G)$ and $D^Q(G)=Tr(G)+D(G)$ are, respectively, the Distance Laplacian and the Distance Signless Laplacian matrices of $G$. The eigenvalues of $D^L(G)$ ( $D^Q(G)$) constitute the Distance Laplacian spectrum ( Distance Signless Laplacian spectrum ). The subdivision graph $S(G)$ of $G$ is obtained by inserting a new vertex into every edge of $G$. We describe here the Distance Spectrum, Distance Laplacian spectrum and Distance Signless Laplacian spectrum of some types of subdivision related graphs of a regular graph in the terms of its adjacency spectrum. We also derive analytic expressions for the distance energy of $\bar{S}(C_p)$, partial complement of the subdivision of a cycle $C_p$ and that of $\overline {S\left( {C_p }\right)}$, complement of the even cycle $C_{2p}$.

For a simple graph G of order n, let $\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu_{n}=0$ be its Laplacian eigenvalues, and let $q_{1}\geq q_{2}\geq\cdots\geq q_{n}\geq0$ be its signless Laplacian eigenvalues. The Laplacian-energy-like invariant and incidence energy of G are defined as, respectively, $$\mathit{LEL}(G)=\sum_{i=1}^{n-1}\sqrt{ \mu_{i}} \quad\mbox{and}\quad \mathit {IE}(G)=\sum_{i=1}^{n} \sqrt{q_{i}}. $$ In this paper, we present some new upper and lower bounds on LEL and IE of line graph, subdivision graph, para-line graph and total graph of a regular graph, some of which improve previously known results. The main tools we use here are the Cauchy-Schwarz inequality and the Ozeki inequality.