Incident Edges Research Articles

Local observables in SUq(2) lattice gauge theory

We consider a deformation of 3D lattice gauge theory in the canonical picture, first classically, based on the Heisenberg double of $\mathrm{SU}(2)$, then at the quantum level. We show that classical spinors can be used to define a fundamental set of local observables. They are invariant quantities that live on the vertices of the lattice and are labeled by pairs of incident edges. Any function on the classical phase space, e.g., Wilson loops, can be rewritten in terms of these observables. At the quantum level, we show that spinors become spinor operators. The quantization of the local observables then requires the use of the quantum $\mathcal{R}$ matrix, which we prove to be equivalent to a specific parallel transport around the vertex. We provide the algebra of the local observables, as a Poisson algebra classically, then as a $q$ deformation of ${\mathfrak{so}}^{*}(2n)$ at the quantum level. This formalism can be relevant to any theory relying on lattice gauge theory techniques such as topological models, loop quantum gravity or of course lattice gauge theory itself.

Modular total vertex irregularity strength of graphs

<abstract><p>A (modular) vertex irregular total labeling of a graph $ G $ of order $ n $ is an assignment of positive integers from $ 1 $ to $ k $ to the vertices and edges of $ G $ with the property that all vertex weights are distinct. The vertex weight of a vertex $ v $ is defined as the sum of numbers assigned to the vertex $ v $ itself and to the edge's incident, while the modular vertex weight is defined as the remainder of the division of the vertex weight by $ n $. The (modular) total vertex irregularity strength of $ G $ is the minimum $ k $ for which such labeling exists. In this paper, we obtain estimations on the modular total vertex irregularity strength, and we evaluate the precise values of this invariant for certain graphs.</p></abstract>

Ramsey and Gallai–Ramsey numbers for the union of paths and stars

Given k graphs H 1 , H 2 , … , H k , the k -color Ramsey number R ( H 1 , H 2 , … , H k ) is defined as the minimum integer n such that every k -edge-coloring of K n contains a monochromatic H i , for some i ∈ [ k ] . If H 1 = H 2 = … = H k = H , then we write the number as R k ( H ) . Given two graphs G and H , and a positive integer k , define the Gallai–Ramsey number gr k ( G : H ) as the minimum number of vertices n such that any exact k -edge coloring of K n contains either a rainbow copy of G or a monochromatic copy of H . Much like Ramsey numbers, Gallai–Ramsey numbers have gained a reputation as being difficult to compute in general. In this paper, we determine the exact values or upper and lower bounds of Gallai–Ramsey number gr k ( G : H ) and Ramsey numbers R 2 ( H ) , R 3 ( H ) when H is the union of a path and a star, and G is a 3-star or a 4-path or P 4 + , where P 4 + is the graph consisting of a P 4 with one extra edge incident with an inner vertex.

On connectivities of edge-colored graphs

Let ( G , c ) be an edge-colored graph where c : E ( G ) → N is an edge-coloring. For x , y ∈ V ( G ) , if there exists a sequence of properly colored cycles C 1 , C 2 , … , C l in G such that x ∈ V ( C 1 ) , y ∈ V ( C l ) and V ( C i ) ∩ V ( C i + 1 ) ≠ 0̸ for any 1 ≤ i < l , then we say the pair x , y is cyclically color-connected , denoted x ≈ y . The color of the edge incident to an end vertex x of a path P is denoted by c x ( P ) . If there exist two properly colored paths P , Q joining x and y such that c x ( P ) ≠ c x ( Q ) and c y ( P ) ≠ c y ( Q ) , then the pair x , y is called color-connected , denoted x ∼ y . If color-connectivity satisfies a transitive relation on V ( G ) , then we say ( G , c ) is convenient . It is straightforward to see that if x ≈ y , then always x ∼ y . However the converse does not hold always. An edge-colored graph ( G , c ) is called strongly convenient if the converse holds. Saad showed that 2-edge-colored complete graphs are convenient. Bang-Jensen and Gutin generalized it for some family of 2-edge-colored complete multipartite graphs. In this paper, we extend those results as follows: any k -edge-colored complete l -partite graph where k , l ∈ N is strongly convenient.

Square Difference Prime Labeling for Duplication of Graphs

Let G(V, E) be a graph with pvertices and qedges. Let f∶ V (G) → {0,1,2,…, p-1} be a bijection such that the induced function f*: E(G) → N defined by f_sqdp* (uv)=|[f(u) ]2-[f(v) ]2 | for everyuv∈E(G).If f_sqdp* is injective, then f_sqdp*is calledsquare difference labeling of G.A graph Gwhich admits square difference labeling is called square difference graph. The greatest common incidence number (gcin) of a vertex v of degree v &gt; 1 is defined as the greatest common divisor (g.c.d) of the labels of the incident edges on v. A square difference labeling fis said to be a square difference prime labeling if for each vertex v of degree &gt;1 then gcin(v) = 1. In this paper we investigate the square difference prime labelling of Petal graph and duplication of petal graph

New Approximation Operators using Combined Edges Systems

In this paper, we introduce new approximation operators such as first lower approximation and first upper approximation of a sub undirected graph ƕ ⊆ Ω by using incident edges system and non-incident edges system respectively. Some properties of these concepts are investigated. In addition the first accuracy of lower and upper approximation operators are introduced and some of its characteristics are studied.

Resistance-distance betweenness: Definition, properties, and application in complex linguistic networks

Betweenness centrality measures how a node lies on the paths between other nodes. We propose the so-called resistance-distance betweenness (RDB) which takes a new viewpoint on scoring paths and contribution of each node in the paths. This new betweenness is developed based on the two facts: (1) the resistance distance characterizes the aggregate distance between two nodes in the context of electrical networks; (2) the resistance distance has an edge-wise decomposition form. Thus, the RDB of a node is defined as the sum of resistance distance components associated with the node’s incident edges, which measures how a node lies on the “aggregated path” between other nodes. The basic properties of RDB and its relation and difference from current-flow betweenness (CFB) are investigated. Then, we apply the RDB to the study of an open question in linguistic networks, i.e., to check whether the Yuan poems written by Han and non-Han poets are similar. We also compare the performance of RDB with CFB. The betweenness analysis of the character co-occurrence networks of Han’s and non-Han’s poems provides a fresh and quantitative perspective to support that Han’s and non-Han’s poems are similar in general.

Graphs with large total angular resolution

The total angular resolution of a straight-line drawing is the minimum angle between two edges of the drawing. It combines two properties contributing to the readability of a drawing: the angular resolution, which is the minimum angle between incident edges, and the crossing resolution, which is the minimum angle between crossing edges. We consider the total angular resolution of a graph, which is the maximum total angular resolution of a straight-line drawing of this graph.We prove tight bounds for the number of edges for graphs for some values of the total angular resolution up to a finite number of well specified exceptions of constant size. In addition, we show that deciding whether a graph has total angular resolution at least 60∘ is NP-hard. Further we present some special graphs and their total angular resolution.

FOREST FRAGMENTATION IN AN UNREGULATED PROTECTED AREA ON THE ATLANTIC COAST OF BRAZIL

Brazilian protected areas are subjected to an intense process of fragmentation due to the lack of environmental regularization and anthropic activities inside them. This study evaluates the forest fragmentation of the Alto Cariri National Park inside the Atlantic Forest of northeastern Brazil. The mapping of forest fragments was performed with the visual interpretation of a satellite image, analyzed based on landscape metrics and physical and topographic conditions. The results show a high forest fragmentation, which is responsible for the segmentation of the forest patch into 120 fragments, predominantly smaller than 5 ha, with an irregular shape and incident edge effect. The obstacles to the preservation of the Atlantic Forest are due to the irregular use of these areas for the production of cattle in pastures and the lack of agility of the public power to implement the expropriation of the population, the drafting of a management plan and decision make.

REFINEMENT OF THE FOUR-COMPONENT SYSTEM MgO – CaO – Al2O3 – Cr2O3 SUB-SOLIDUS STRUCTURE INCLUDING THE Сa6Al4Cr2O15 COMPOUND

The results of theoretical calculations of the subsolidus structure of the four-component system MgO – CaO – Al2O3 – Cr2O3 are presented, taking into account the three-component compound Сa6Al4Cr2O15. Thermodynamic and geometric topological analysis methods were used to establish stable conodes in the specified system. When dividing the concentration tetrahedron MgO – CaO – Al2O3 – Cr2O3 into elementary ones with the involvement of a geometro-topological analysis, it is possible to make an unambiguous closure to the elementary tetrahedron of faces with incident edges (connodes) through a common vertex. The use of this method allows to minimize the number of necessary thermodynamic calculations. The tetrahedra of the MgO – CaO – Al2O3 – Cr2O3 system, taking into account the stable phases, is caused by the presence of three "internal" conodes passing through the three-dimensional space of the concentration tetrahedron: Сa6Al4Cr2O15 – MgO; MgCa3Al4O10 – CaCr2O4, CaCr2O4 – MgAl2O4, which determine the presence of 14 elementary tetrahedra in the subsolidus region. The lengths of conodes and volumes of elementary tetrahedra were calculated taking into account barycentric coordinates and elements of Euclidean geometry. The phases included in the composition of the tetrahedron with the largest relative volume and the smallest asymmetry CaCr2O4 – СaAl2O4 –MgAl2O4 – MgO (185.6 ‰ and 2.55, respectively) have the highest probability of existence in the system, which will allow the development of a sustainable technology for creating composite materials based on calcium aluminochromite cement, with periclase as an aggregate, without special techniques dosage of initial components. At the same time, in the event of a violation of the technological process, magnesium aluminate spinel will be synthesized in the composition of the unformed material, which will not lead to deterioration of the performance characteristics of the composite used. It should be noted that elementary tetrahedra with a high degree of asymmetry, which require appropriate dosage accuracy when predicting the phase composition of synthesized composite materials, are located in areas of the system that are not technologically advantageous from the point of view of obtaining unformed refractories based on special binder materials, and are interest for materials scientists in the ceramic and refractory industries.

The weighted sitting closer to friends than enemies problem in the line

The weighted Sitting Closer to Friends than Enemies (SCFE) problem is to find an injection of the vertex set of a given weighted graph into a given metric space so that, for every pair of incident edges with different weight, the end vertices of the heavier edge are closer than the end vertices of the lighter edge. The Seriation problem is to find a simultaneous reordering of the rows and columns of a symmetric matrix such that the entries are monotone nondecreasing in rows and columns when moving towards the diagonal. If such a reordering exists, it is called a Robinson ordering. In this work, we establish a connection between the SCFE problem and the Seriation problem. We show that if the extended adjacency matrix of a given weighted graph G has no Robinson ordering then G has no injection in $$\mathbb {R}$$ that solves the SCFE problem. On the other hand, if the extended adjacency matrix of G has a Robinson ordering, we construct a polyhedron that is not empty if and only if there is an injection of the vertex set of G in $$\mathbb {R}$$ that solves the SCFE problem. As a consequence of these results, we conclude that deciding the existence of (and constructing) such an injection in $$\mathbb {R}$$ for a given complete weighted graph can be done in polynomial time. On the other hand, we show that deciding if an incomplete weighted graph has such an injection in $$\mathbb {R}$$ is NP-Complete.

Double Vizing fans in critical class two graphs

Let G $G$ be a simple graph and χ ′ ( G ) $\chi ^{\prime} (G)$ be the chromatic index of G $G$ . We call G $G$ a Δ ${\rm{\Delta }}$ -critical graph if χ ′ ( G − e ) = χ ′ ( G ) − 1 = Δ $\chi ^{\prime} (G-e)=\chi ^{\prime} (G)-1={\rm{\Delta }}$ for every edge e $e$ of G $G$ , where Δ ${\rm{\Delta }}$ is maximum degree of G $G$ . Let e = x y $e=xy$ be an edge of Δ ${\rm{\Delta }}$ -critical graph G $G$ and φ $\varphi $ be an (proper) edge Δ ${\rm{\Delta }}$ -coloring of G − e $G-e$ . An e-fan is a sequence F e = ( x , e , y , e 1 , z 1 , … , e p , z p ) ${F}^{e}=(x,e,y,{e}_{1},{z}_{1},\ldots ,{e}_{p},{z}_{p})$ of alternating vertices and distinct edges such that edge e i ${e}_{i}$ is incident with x $x$ or y $y$ , z i ${z}_{i}$ is another endvertex of e i ${e}_{i}$ and φ ( e i ) $\varphi ({e}_{i})$ is missing at a vertex before z i ${z}_{i}$ for each i $i$ with 1 ≤ i ≤ p $1\le i\le p$ . In this paper, we prove that if min { d ( x ) , d ( y ) } ≤ Δ − 1 $\min \{d(x),d(y)\}\le {\rm{\Delta }}-1$ , where d ( x ) $d(x)$ and d ( y ) $d(y)$ denote the degrees of vertices x $x$ and y $y$ , respectively, then colors missing at different vertices of V ( F e ) $V({F}^{e})$ are distinct. Clearly, a Vizing fan is an e $e$ -fan with the restricting that all edges e i ${e}_{i}$ being incident with one fixed endvertex of edge e $e$ . This result gives a common generalization of several recently developed new results on multifan, double fan, Kierstead path of four vertices, and broom. By treating some colors of edges incident with vertices of low degrees as missing colors, Kostochka and Stiebitz introduced C $C$ -fan. In this paper, we also generalize the C $C$ -fan from centered at one vertex to one edge.

Improved approximation for maximum edge colouring problem

The anti-Ramsey number , a r ( G , H ) is the minimum integer k such that in any edge colouring of G with k colours there is a rainbow subgraph isomorphic to H , namely, a copy of H with each of its edges assigned a different colour. The notion was introduced by Erdös and Simonovits in 1973. Since then the parameter has been studied extensively. The case when H is a star graph was considered by several graph theorists from the combinatorial point of view. Recently this case received the attention of researchers from the algorithm community because of its applications in interface modelling of wireless networks. To the algorithm community, the problem is known as maximum edge q -colouring problem: Find a colouring of the edges of G , maximizing the number of colours satisfying the constraint that each vertex spans at most q colours on its incident edges. It is easy to see that the maximum value of the above optimization problem equals a r ( G , K 1 , q + 1 ) − 1 . In this paper, we study the maximum edge 2-colouring problem from the approximation algorithm point of view. The case q = 2 is particularly interesting due to its application in real-life problems. Algorithmically, this problem is known to be NP-hard for q ≥ 2 . For the case of q = 2 , it is also known that no polynomial-time algorithm can approximate to a factor less than 3 / 2 assuming the unique games conjecture. Feng et al. showed a 2-approximation algorithm for this problem. Later Adamaszek and Popa presented a 5 / 3 -approximation algorithm with the additional assumption that the input graph has a perfect matching. Note that the obvious but the only known algorithm issues different colours to the edges of a maximum matching (say M ) and different colours to the connected components of G ∖ M . In this article, we give a new analysis of the aforementioned algorithm to show that for triangle-free graphs with perfect matching the approximation ratio is 8 / 5 . We also show that this algorithm cannot achieve a factor better than 58 / 37 on triangle free graphs that has a perfect matching. The contribution of the paper is a completely new, deeper and closer analysis of how the optimum achieves a higher number of colours than the matching based algorithm, mentioned above.

4-total edge product cordial for some star related graphs

&lt;p&gt;Let G = (V (G), E(G)) be a graph, define an edge labeling function ψ from E(G) to {0, 1, . . . , k − 1} where k is an integer, 2 ≤ k ≤ |E(G)|, induces a vertex labeling function ψ∗ from V (G) to {0, 1, . . . , k − 1} such that ψ∗(v) = ψ(e1) × ψ(e2) × . . . × ψ(en) mod k where e1, e2, . . . , en are all edge incident to v. This function ψ is called a k-total edge product cordial (or simply k-TEPC) labeling of G if the absolute difference between number of vertices and edges labeling with i and number of vertices and edges labeling with j no more than 1 for all i, j ∈ {0, 1, . . . , k − 1}. In this paper, 4-total edge product cordial labeling for some star related graphs are determined.&lt;/p&gt;

Properly colored 2-factors of edge-colored complete bipartite graphs

Let Gc be an edge-colored graph. The minimum color degree of Gc, denoted δc(G), is the largest integer k such that for every vertex v, there are at least k distinct colors on edges incident with v. A cycle in Gc is called properly colored if any of its adjacent edges have distinct colors. A properly colored 2-factor of Gc is a spanning subgraph consisting of vertex-disjoint properly colored cycles of Gc. It was conjectured in the literature that, if δc(Km,n)≥m+n4+1, then each vertex of Km,nc is contained in a properly colored cycle of length l for any even integer l with 4≤l≤min⁡{2m,2n}. In this paper, we consider the existence of properly colored 2-factor in edge-colored bipartite graphs and obtain the following two results: (i) if δc(Kn,n)>3n/4, then Kn,nc contains a properly colored 2-factor. (ii) For every number ε with 0<ε<1/2, there exists an integer n0(ε) such that every Kn,nc with n≥n0(ε) and δc(Kn,n)≥(1/2+ε)n contains a properly colored 2-factor.

A Characterization for the Neighbor-Distinguishing Index of Planar Graphs

Symmetry, such as structural symmetry, color symmetry and so on, plays an important role in graph coloring. In this paper, we use structural symmetry and color symmetry to study the characterization for the neighbor-distinguishing index of planar graphs. Let G be a simple graph with no isolated edges. The neighbor-distinguishing edge coloring of G is a proper edge coloring of G such that any two adjacent vertices admit different sets consisting of the colors of their incident edges. The neighbor-distinguishing index χa′(G) of G is the smallest number of colors in such an edge coloring of G. It was conjectured that if G is a connected graph with at least three vertices and G≠C5, then χa′(G)≤Δ+2. In this paper, we show that if G is a planar graph with maximum degree Δ≥13, then Δ≤χa′(G)≤Δ+1, and, further, χa′(G)=Δ+1 if and only if G contains two adjacent vertices of maximum degree.

Succinct navigational oracles for families of intersection graphs on a circle

We consider the problem of designing succinct navigational oracles, i.e., succinct data structures supporting basic navigational queries such as degree, adjacency and neighborhood efficiently for intersection graphs on a circle, which include graph classes such as circle graphs, k-polygon-circle graphs, circle-trapezoid graphs, trapezoid graphs. The degree query reports the number of incident edges to a given vertex, the adjacency query asks if there is an edge between two given vertices, and the neighborhood query enumerates all the neighbors of a given vertex. We first prove a general lower bound for these intersection graph classes, and then present a uniform approach that lets us obtain matching lower and upper bounds for representing each of these graph classes. More specifically, our lower bound proofs use a unified technique to produce tight bounds for all these classes, and this is followed by our data structures which are also obtained from a unified representation method to achieve succinctness for each class. In addition, we prove a lower bound of space for representing trapezoid graphs, and give a succinct navigational oracle for this class of graphs.

Antimagic Labeling for Product of Regular Graphs

An antimagic labeling of a graph G=(V,E) is a bijection from the set of edges of G to 1,2,⋯,E(G) and such that any two vertices of G have distinct vertex sums where the vertex sum of a vertex v in V(G) is nothing but the sum of all the incident edge labeling of G. In this paper, we discussed the antimagicness of rooted product and corona product of graphs. We proved that if we let G be a connected t-regular graph and H be a connected k-regular graph, then the rooted product of graph G and H admits antimagic labeling if t≥k. Moreover, we proved that if we let G be a connected t-regular graph and H be a connected k-regular graph, then the corona product of graph G and H admits antimagic labeling for all t,k≥2.

Topology of clique complexes of line graphs

The clique complex of a graph G is a simplicial complex whose simplices are all the cliques of G, and the line graph L(G) of G is a graph whose vertices are the edges of G and the edges of L(G) are incident edges of G. In this article, we determine the homotopy type of the clique complexes of line graphs for several classes of graphs including triangle-free graphs, chordal graphs, complete multipartite graphs, wheel-free graphs, and 4-regular circulant graphs. We also give a closed form formula for the homotopy type of these complexes in several cases.

On a vertex-capturing game

In this paper, we study the recently introduced scoring game played on graphs called the Edge-Balanced Index Game. This game is played on a graph by two players, Alice and Bob, who take turns colouring an uncoloured edge of the graph. Alice plays first and colours edges red, while Bob colours edges blue. The game ends once all the edges have been coloured. A player captures a vertex if more than half of its incident edges are coloured by that player, and the player that captures the most vertices wins.Using classical arguments from the field, we first prove general properties of this game. Namely, we prove that there is no graph in which Bob can win (if Alice plays optimally), while Alice can never capture more than 2 more vertices than Bob (if Bob plays optimally). Through dedicated arguments, we then investigate more specific properties of the game, and focus on its outcome when played in particular graph classes. Specifically, we determine the outcome of the game in paths, cycles, complete bipartite graphs, and Cartesian grids, and give partial results for trees and complete graphs.