Some snake graphs are edge odd graceful

An edge odd graceful labeling of graph G with q number of edges is a bijection f : E(G) → {1,3,5, …,2q - 1} so that induced mapping f+ : V(G) → {0, 1,2, … ,2q - 1} given by f+(x) = ExyEE(G) f(xy) (mod 2q) is injective. A triangular snake net graph (m - 1)C3m is graph obtained from a path u1 u2 u3 … um + 1 by joining every ui and ui + 1 to a new vertex vi with 1 ≤ i ≤ m and joining every vi to vi+1 with 1 ≤ i ≤ m - 1. A quadrilateral snake net graph C4,4m is a graph obtained from vertices u1, u2, u3, …, u2m + 1 by joining every ui and ui + 1 to two vertices vi and v2i-1with 1 ≤ i ≤ 2m, and joining every v4i - 2 and v4i to a new vertex w2i and joining every v4i - 3 and v4i-1 to a new vertex w2i - 1 with 1 ≤ i ≤ m. An alternate quadrilateral snake net A(C4,4m) is graph obtained from vertices u1, u2, u3, … , u3m by joining every u3i and u3i + 1 with 1 ≤ i ≤ m - 1, and joining u3i - 2 and u3i - 1 to two new vertices v4i-3 and v4i-2, joining u3i-1 and u3i to two new vertices v4i-1 and v4i, joining v4i-3 and v4i-1 to a new vertex w2i-1, and joining v4i-2 and v4i to two a new vertex w2i with 1 ≤ i ≤ m. A quadrilateral pleated snake graph C42 is a graph obtained from vertices u0, u, and v by joining every u0 and u to new vertices ui and joining every u0 and v to new vertices vi with 1 ≤ i ≤ 2 m + 2. In this paper, we study edge odd graceful labeling for the following families of snake graphs such as triangular snake net, quadrilateral snake net, alternate quadrilateral snake net, and quadrilateral pleated snake.

Given a simple connected undirected graph G and let k be the maximum number of its vertices and its edges. Let f be a bijective labeling from the set of its edges to the set of odd integers from 1 up to 2q − 1, where q is the number of edges of G. The labeling f is called an edge odd graceful labeling on G if the weights of any two different vertices are different, where the weight of a vertex v is defined as the sum mod(2k) of all labels of edges that are incident to v. A graph is called an edge odd graceful graph if it admits an edge odd graceful labeling. In this paper, we show that there are some new classes of graphs that are edge odd graceful.

Networks derived from the hypercube are noted for their efficiency in simulating any bounded-degree communication network and hence they play a significant role in parallel machines. For a graph [Formula: see text], a geodetic cover is a subset [Formula: see text] of [Formula: see text], satisfying the condition that every vertex of [Formula: see text] other than the vertices in [Formula: see text] are covered using geodesics connecting any two vertices in [Formula: see text]. The geodetic number of [Formula: see text] is the least order of such covers in [Formula: see text]. This seminal distance-based parameter is well employed in the areas of location theory, convexity theory and game theory. The strong variant and its edge version are recent variations of the problem that were introduced in due of their applications in social networks. The most established and strong hypercube derivative networks are studied in this paper with respect to certain geodetic parameters. Notably, among the networks that we have considered, the butterfly and the Benes networks belong to the class of bipartite graphs for which the geodetic number problem and its strong version are NP-complete. We have determined the geodetic parameters that include the geodetic number, the strong geodetic number and their corresponding edge variants of the networks taken into consideration. In the process, we have identified a new class of graphs, as semi-extreme edge geodesic graphs and that the enhanced butterfly networks belong to this class. The study of such geodetic parameters in the structural models of parallel computation could be useful for the design of efficient algorithms for the structures considered.

The metric representation of a vertex v of a graph G is a finite vector representing distances of v with respect to vertices of some ordered subset S⊆V (G). If no suitable subset of S provides separate representations for each vertex of V(G), then the set S is referred to as a minimal resolving set. The metric dimension of G is the cardinality of the smallest (with respect to its cardinality) minimal resolving set. A resolving set S is secure if for any v∈V–S, there exists x∈S such that (S–{x})∪{v} is a resolving set. For various classes of graphs, the value of the secure resolving number is determined and defined. The secure metric dimension of the graph classes is being studied in this work. The results show that different graph families have different metric dimensions.

Matthews and Sumner have proved in [10] that if G is a 2-connected claw-free graph of order n such that δ(G) ≥ (n-2) /3, then G is Hamiltonian. We say that a graph is almost claw-free if for every vertex v of G, 〈N(v)〉 is 2-dominated and the set A of centers of claws of G is an independent set. Broersma et al. [5] have proved that if G is a 2-connected almost claw-free graph of order n such that n such that δ(G) ≥ (n-2) /3, then G is Hamiltonian. We generalize these results by considering the graphs satisfying the following property: for every vertex v ∈ A , there exist exactly two vertices x and y of V\A such that N(v) ⊆ N[x] ∪ N[y] . We extend some other known results on claw-free graphs to this new class of graphs.

We construct the first known examples of infinite subgroups of the outer automorphism group of Out(A_Gamma), for certain right-angled Artin groups A_Gamma. This is achieved by introducing a new class of graphs, called focused graphs, whose properties allow us to exhibit (infinite) projective linear groups as subgroups of Out(Out(A_Gamma)). This demonstrates a marked departure from the known behavior of Out(Out(A_Gamma)) when A_Gamma is free or free abelian, as in these cases Out(Out(A_Gamma)) has order at most 4. We also disprove a previous conjecture of the second author, producing new examples of finite order members of certain Out(Aut(A_Gamma)).

A Semidefinite Programming (SDP) relaxation is an effective computational method to solve a Sensor Network Localization problem, which attempts to determine the locations of a group of sensors given the distances between some of them. In this paper, we analyze and determine new sufficient conditions and formulations that guarantee that the SDP relaxation is exact, i.e., gives the correct solution. These conditions can be useful for designing sensor networks and managing connectivities in practice. Our main contribution is threefold: First, we present the first non-asymptotic bound on the connectivity (or radio) range requirement of randomly distributed sensors in order to ensure the network is uniquely localizable with high probability. Determining this range is a key component in the design of sensor networks, and we provide a result that leads to a correct localization of each sensor, for any number of sensors. Second, we introduce a new class of graphs that can always be correctly localized by an SDP relaxation. Specifically, we show that adding a simple objective function to the SDP relaxation model will ensure that the solution is correct when applied to a triangulation graph. Since triangulation graphs are very sparse, this is informationally efficient, requiring an almost minimal amount of distance information. Finally, we analyze a number of objective functions for the SDP relaxation to solve the localization problem for a general graph.Key wordsSensor network localizationGraph realizationSemidefinite programmingSubject Classifications90C2290C4690C9090B18

Polar graphs and maximal independent sets

In 1983, Klaus studied a class of potentials with bumps and computed the essential spectrum of the associated Schr{\"o}dinger operator with the help of some localisations at infinity. A key hypothesis is that the distance between two consecutive bumps tends to infinity at infinity. In this article, we introduce a new class of graphs (with patterns) that mimics this situation, in the sense that the distance between two patterns tends to infinity at infinity. These patterns tend, in some way, to asymptotic graphs. They are the localisations at infinity. Our result is that the essential spectrum of the Laplacian acting on our graph is given by the union of the spectra of the Laplacian acting on the asymptotic graphs. We also discuss the question of the stability of the essential spectrum in the appendix.

For a simple graph G on n vertices and an integer k with 1 ⩽ k ⩽ n, denote by $$\mathcal{S}^+_k$$ (G) the sum of k largest signless Laplacian eigenvalues of G. It was conjectured that $$\mathcal{S}^+_k(G)\leqslant{e}(G)+(^{k+1}_{\;\;2})$$ (G) ⩽ e(G) + (k+1 2), where e(G) is the number of edges of G. This conjecture has been proved to be true for all graphs when k ∈ {1, 2, n − 1, n}, and for trees, unicyclic graphs, bicyclic graphs and regular graphs (for all k). In this note, this conjecture is proved to be true for all graphs when k = n − 2, and for some new classes of graphs.

Our work becomes integrated into the general problem of the stability of the network ad hoc. Some, works attacked (affected) this problem. Among these works, we find the modelling of the network ad hoc in the form of a graph. We can resume the problem of coherence of the network ad hoc of a problem of allocation of frequency We study a new class of graphs, the fat-extended P4 graphs, and we give a polynomial time algorithm to calculate the Grundy number of the graphs in this class. This result implies that the Grundy number can be found in polynomial time for many graphs

Known realizations of geometric representations of graphs, like contact, intersection, etc., are “continuous”, in the sense that the geometric objects are drawn in Euclidean space with real numbers as coordinates. In this paper, we initiate the study of dicrete versions of contact and intersection graphs and examine their relation to their continuous counterparts. The classes of graphs arising appear to have interesting properties and are thus interesting in their own right. We also study realizability, characterizations as well as intractability questions for the resulting new classes of graphs.1980 Mathematics Subject Classification68111068U05CR CategoriesF.2.2Key Words and PhrasesCoinContactIntersectionInterval graphsDiscretePlanar graphsNP

<abstract><p>Recently, Çanakçi and Schroll proved that associated with a string module $ M(w) $ there is an appropriated snake graph $ \mathscr{G} $. They established a bijection between the corresponding perfect matching lattice $ \mathscr{L}(\mathscr{G}) $ of $ \mathscr{G} $ and the canonical submodule lattice $ \mathscr{L}(M(w)) $ of $ M(w) $. We introduce Brauer configurations whose polygons are defined by snake graphs in line with these results. The developed techniques allow defining snake graphs, which after suitable procedures, build Kronecker modules. We compute the dimension of the Brauer configuration algebras and their centers arising from the different processes. As an application, we estimate the trace norm of the canonical non-regular Kronecker modules and some families of trees associated with some snake graphs classes.</p></abstract>

(G);E(G)) with p vertices and q edges, if there exists an injectionfunction : V (G) ! f0; 1; 2; ; 2q ô€€€ 1g with each edge uv assignedthe label jf(u)ô€€€f(v)j, the resulting edge labels are f1; 3; 5; ; 2qô€€€1g.The tensor product of two graphs, G and H, has a vertex setV (G) V (H) and an edge between (u; v) and (u0; v0), i both uu0 2 E(G) andvv0 2 E(H), here we denote tensor product by ^. In this paper, weprove Sn ^ Sm, Sn ^ Pm and Cn ^ Sm are odd graceful.

On weakly diamond-free Berge graphs