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International audience; We consider adaptations of the Mumford-Shah functional to graphs. These are based on discretizations of nonlocal approximations to the Mumford-Shah functional. Motivated by applications in machine learning we study the random geometric graphs associated to random samples of a measure. We establish the conditions on the graph constructions under which the minimizers of graph Mumford-Shah functionals converge to a minimizer of a continuum Mumford-Shah functional. Furthermore we explicitly identify the limiting functional. Moreover we describe an efficient algorithm for computing the approximate minimizers of the graph Mumford-Shah functional.

This paper presents an approach to study initial-boundary value (IBV) problems for integrable nonlinear differential-difference equations (DDEs) posed on a graph. As an illustrative example, we consider the Ablowitz–Ladik system posed on a graph that is constituted by N semi-infinite lattices (edges) connected through some boundary conditions. We first show that analyzing this problem is equivalent to analyzing a certain matrix IBV problem; then we employ the unified transform method (UTM) to analyze this matrix IBV problem. We also compare our results with some previously known studies. In particular, we show that the inverse scattering method (ISM) for the integrable DDEs on the set of integers can be recovered from the UTM applied to our N = 2 graph problem as a particular case, and the non-local reductions of integrable DDEs can be obtained as local reductions from our results.

We study several fractal properties of the Weierstrass-type function where is a cookie cutter map with possibly fractal repeller, and λ and g are functions with proper regularity. In the first part, we determine the box dimension of the graph of W and Hausdorff dimension of its randomised version. In the second part, the Hausdorff spectrum of the local Holder exponent is characterised in terms of thermodynamic formalism. Furthermore, in the randomised case, a novel formula for the lifted Hausdorff spectrum on the graph is provided.

Graph splicing system is a notion originally used to illustrate the one-dimensional string of DNA splicing in the form of a graph. A graph splicing system is associated with a graph splicing scheme where graph splicing rules are defined. A graph splicing rule restricts the possible cuts to occur on the edges of the initial graph(s) in a graph splicing system. The subgraphs of the initial graph are used in the splicing rules to determine the edges that will be cut from the initial graph. The concept of graph splicing system can be applied on various types of graph, hence generates components of spliced graphs depending on the types of the graph splicing rules used. There is a graph splicing rule called as a cutting rule which can be applied on both linear graphs and circular graphs where the graphs are transformed into Pseudo-Linear Form. However, this cutting rule has limited various possible cuts that can occur on the complete graph. Therefore, in this research, the original concept of graph splicing system is applied on complete graphs as the initial graphs. Also, graph splicing rules involving subgraphs of the initial graphs which are also complete graphs, are considered and applied in the graph splicing system. Furthermore, the generated spliced graphs are obtained through the graph splicing system.

An anti-magic labeling of a finite simple undirected graph G is a bijection from the set of edges E(G) to the set of integers {1, 2,…, |E(G)|} such that the vertex sums are pairwise distinct, where the vertex sum at one vertex is the sum of labels of all edges incident to such vertex. A graph is called anti-magic if it admits an anti-magic labeling. In this paper, we established an anti-magic labeling of Card House Graphs.

The article discusses possible approaches to data collection and information collection in multi-level distributed systems. It is suggested to use model based approach. The requirements to these models are analyzed and the system of models is proposed. The system of models describes the structure and behavior of the observed system in terms of graphs. The resource graph, the data flow graph, the control flow graph, and the query flow graph are used. These graphs are automatically constructed on the basis of information received from the observed system in the form of log files. A virtual machine that implements this approach is proposed.

We recall the construction of the Kontsevich graph orientation morphism $\gamma \mapsto {\rm O\vec{r}}(\gamma)$ which maps cocycles $\gamma$ in the non-oriented graph complex to infinitesimal symmetries $\dot{\mathcal{P}} = {\rm O\vec{r}}(\gamma)(\mathcal{P})$ of Poisson bi-vectors on affine manifolds. We reveal in particular why there always exists a factorization of the Poisson cocycle condition $[\![\mathcal{P},{\rm O\vec{r}}(\gamma)(\mathcal{P})]\!] \doteq 0$ through the differential consequences of the Jacobi identity $[\![\mathcal{P},\mathcal{P}]\!]=0$ for Poisson bi-vectors $\mathcal{P}$. To illustrate the reasoning, we use the Kontsevich tetrahedral flow $\dot{\mathcal{P}} = {\rm O\vec{r}}(\gamma_3)(\mathcal{P})$, as well as the flow produced from the Kontsevich--Willwacher pentagon-wheel cocycle $\gamma_5$ and the new flow obtained from the heptagon-wheel cocycle $\gamma_7$ in the unoriented graph complex., Comment: 12 pages. Talk given by R.B. at Group32 (Jul 9--13, 2018; CVUT Prague, Czech Republic). Big formula in Appendix A retained from the (unpublished) Appendix in arXiv:1712.05259 [math-ph]. Signs corrected in v2

Expander graphs are highly connected graphs that have numerous applications in statistical physics, pure mathematics and in computer science. The increased connectivity in expanders are useful to model connections between interconnecting systems which can be considered as a graph composed of particles as vertices and edges represent interactions. This paper focuses on the fascinating and highly active area of research on expander graphs. In this article, different classes of expander graphs such as Schreier graphs, Ramanujan graphs and Lp-expanders are categorized and various constructions of an explicit family of expanders are explored. Based on their construction the chromatic number of these are graphs are obtained.

For a simple connected graphG, letc:V(G) → ℕ be a vertex coloring ofG,where adjacent vertices may be colored the same. The neighborhood color set of a vertexv, denoted byNC(v), is the set of colors of the neighbors ofv. The coloringcis called aset coloringprovided thatNC(u) ≠NC(v) for every pair of adjacent verticesuandvofG. The minimum number of colors needed for a set coloring ofGis referred to as the setchromatic numberofGand is denoted byχs(G).In this work, the set chromatic number of graphs is studied in relation to the graph operation called middle graph. Our results include the exact set chromatic numbers of the middle graph of cycles, paths, star graphs, double-star graphs, and some trees of height 2. Moreover, we establish the sharpness of some bounds on the set chromatic number of general graphs obtained using this operation. Finally, we develop an algorithm for constructing an optimal set coloring of the middle graph of trees of height 2 under some assumptions.

A graph is SSP (Super Strongly Perfect) if all of its (induced) subgraph H in G obsesses a (minimal dominating set) MDS that link up all of its cliques (maximal) in H. It is offered the cyclic structure of k-sun flower graph, friendship and Jahangir graphs and later explored their SSP parameters like counting of colourability, MDS (minimal dominating set) and cliques (maximal) of these graphs.

The set D ⊆ V(G) is called dominating set on graph G so that every vertex not in D is adjacent to at least one vertex in D. The set Dt ⊆ V(G) is called total dominating set on graph G so that the vertex in Dt are neighboring at least one dot in Dt . The smallest cardinality of the total dominating set is referred to as total domination number. The total domination number in G is shown by γt (G). The set of vertex Dt ⊆ V(G) is resolving total dominating set from G if the vertex representation u,υ ∈ V(G) with respect to x ∈ Dt is r(υ|Dt ) so that r(υ|Dt ) ≠ r(u|Dt ). The smallest cardinality of Resolving total dominating set in G is shown by γrt (G). In this article, we provide the results of the study for the differentiating number of total dominance from sunlet graphs.

For any proper set S of V in a graph G(V, E), the S-eccentricity, eG, S (υ) (in short eS (υ)) of a vertex υ in G is max x ∈ S ( d ( υ , x ) ) . The S-center of G is CS (G) = {υ ∈ V | eS (x) ≤ e S(x)∀x ∈ V} and S 1-eccentricity, eG,S1 (v) (in short eS1 (V)) of a vertex v in S is max x ∈ V = S ( d ( υ , x ) ) . S 1-center of G is CS1 (G) = {υ ∈ V | eS1 (x) ≤ e S1 (x)∀x ∈ V}. Then G is called a center-smooth graph if CS (G) = CS1 (G) and the set S is defined to be a center-smooth set. We identify the center smooth sets of certain classes of graphs namely, Kq,p , Kp – q, Kp , wheel graphs and lollipop graph and enumerate them for many of these graph classes. We also introduce the concept of center smooth number, which is defined as the number of distinct center smooth set of a graph G, and determine the center smooth number of some graph classes.

The slope number is defined to be the minimum number of slopes required to draw the graph. In the present paper, investigation on slope number of Jahangir graph is focused and studied elaborately, since the defined graph has an excellent application on transmitting confidential information between nuclear sites. In this paper, slope number of Jahangir graph J2,m is determined and studied elaborately. This is an optimization problem and is NP-hard to determine for any arbitrary graph.

For a connected graph G with a distribution of p pebbles on it, a pebbling move involves the removal of two pebbles from a vertex and an addition of one pebble to an adjacent vertex. The cover pebbling number γ(G) is the minimum number of pebbles required to place one pebble on each vertex of the graph G after a sequence of pebbling moves, regardless of the initial configuration of the pebbles. It is known that the problem of computing the cover pebbling number is NP-complete [7]. In this paper we compute the cover pebbling number of Comb, Friendship and Helm graphs.

A graph G is supposed to be trivially perfect if, in each induced subgraph H of G, the number of maximal cliques in H equivalents to the size of a maximum independent set in H. Trivially perfect graphs is subclasses of notable perfect graphs and its characterization have numerous continuous applications and it is adequate to research its subclasses. Along with this idea, in this paper, it is discussed trivially perfect graphs on the windmill graph and demonstrated a few outcomes on trivially perfect graphs.

Labelling of graphs has been the allocation of integers to the nodes or edges, or perhaps both, liable to revision. Graph coloring is indeed a specialized version of graph labelling, which is an allocation of labels consists of assessment to as colors to graph that are subject to several limitations. From its simplified sense it is indeed a method to color the nodes of a graph. This article deals with the harmonious coloring of central graph of some Mesh derived Architectures. A coloring is a harmonious coloring if it is a proper coloring with the intension of a couple of colors emerges towards one edge at the most.

Let $G$ be a nontrivial edge-colored connected graph. An edge-cut $R$ of $G$ is called a {\it rainbow edge-cut} if no two edges of $R$ are colored with the same color. For two distinct vertices $u$ and $v$ of $G$, if an edge-cut separates them, then the edge-cut is called a {\it $u$-$v$-edge-cut}. An edge-colored graph $G$ is called \emph{strong rainbow disconnected} if for every two distinct vertices $u$ and $v$ of $G$, there exists a both rainbow and minimum $u$-$v$-edge-cut ({\it rainbow minimum $u$-$v$-edge-cut} for short) in $G$, separating them, and this edge-coloring is called a {\it strong rainbow disconnection coloring} (srd-{\it coloring} for short) of $G$. For a connected graph $G$, the \emph{strong rainbow disconnection number} (srd-{\it number} for short) of $G$, denoted by $\textnormal{srd}(G)$, is the smallest number of colors that are needed in order to make $G$ strong rainbow disconnected. In this paper, we first characterize the graphs with $m$ edges such that $\textnormal{srd}(G)=k$ for each $k \in \{1,2,m\}$, respectively, and we also show that the srd-number of a nontrivial connected graph $G$ equals the maximum srd-number among the blocks of $G$. Secondly, we study the srd-numbers for the complete $k$-partite graphs, $k$-edge-connected $k$-regular graphs and grid graphs. Finally, we show that for a connected graph $G$, to compute $\textnormal{srd}(G)$ is NP-hard. In particular, we show that it is already NP-complete to decide if $\textnormal{srd}(G)=3$ for a connected cubic graph. Moreover, we show that for a given edge-colored (with an unbounded number of colors) connected graph $G$ it is NP-complete to decide whether $G$ is strong rainbow disconnected., 16 pages, 1 figure

The labeling is said to be lucky labeling of the graph if the vertices of the graph are labeled by natural number with satisfying the condition that sum of labels over the adjacent of the vertices in the graph are not equal and if vertices are isolated vertex then the sum of the vertex is zero. The least natural number which labelled the graph is the lucky number. The Lucky Number of graph G is denoted by η(G). The labeling defined as proper labeling if the vertices of the graph are labeled by natural number with fulfilling the condition that label of adjacent vertices is not the same. The labeling is defined as proper lucky labeling if labeling is proper and also lucky. The proper lucky number of graph G is denoted by η p (G). Here we obtain a lucky number and proper lucky number for family of quadrilateral snake graph such as quadrilateral snake graph, double quadrilateral snake graph, alternate quadrilateral snake graph and double alternate quadrilateral snake graph.

In crisp graphs, if a graph G is given then we can define line graph L(G) of it. In a similar way, for a given fuzzy graph, we can determine its fuzzy line graph. A number of important results about fuzzy line graphs have been obtained. However, study of anti fuzzy line graphs is still open. Motivated by this, we are interested in studying anti fuzzy line graphs. In this paper, we define the concept of anti fuzzy line graphs. We derive a necessary and sufficient condition for an anti fuzzy graph to be isomorphic to its corresponding anti fuzzy line graph. We investigate when an isomorphism between two anti fuzzy graphs follows from an isomorphism of their corresponding anti fuzzy line graphs. We conjecture a necessary and sufficient condition for an anti fuzzy graph to be the anti fuzzy line graph of some anti fuzzy graph. In particular, we give some result related to regular anti fuzzy line graphs.

This paper presents preparations, execution and reflection on results for the physics part of a CSI task for the European Union Science Olympiad 2010, taking place at the University of Gothenburg. The participants were required to make a graph of the cooling of a swede (Brassica napus), as a proxy for the cooling of a murder victim, and to choose a mathematical expression for the cooling curve, as well as assigning an approximate time of death.

Let G(V, E) be a finite, simple and undirected graph with a vertex set V and an edge set E. An edge irregular total k-labeling is a map f : V ⋃ E → {1, 2, …, k} such that for any two different edges xy and x′y′ in E, ω(xy) ≠ ω(x′y′) where ω(xy) = f(x) + f(y) + f(xy). The minimum k for which the graph G admits an edge irregular total k-labeling is called the total edge irregularity strength of G, denoted by tes(G). We have constructed the formula of an edge irregular total k-labelling and determined the total edge irregularity strength of book graphs and double book graphs. In this paper, we construct an edge irregular total k-labeling that can be used for book graphs, double book graphs, and triple book graphs. We also show the exact value of the total edge irregularity strength of triple book graphs.

This paper introduces a general question answering (QA) framework for Chinese domain knowledge graphs to serve various industries. The question is analyzed by shielding invalid characters in this framework, and the TransE calculation of entities and relations is used to search answer, rather than traditional logical queries that rely heavily on predefined rules. In this way, the QA framework can be easily applied to different fields with little manual participation, and reduce the migration cost caused by the lack of scalability of the existing QA systems, which are designed only for specific domains. Then, the proposed framework is operated on the space launch knowledge graph and has achieved HITS@1, HITS@3, HITS@5 scores of 60.60%, 80.87%, 85.61% separately, which is competitive with other restricted domain knowledge graphs. In addition, numerical results on the knowledge graph of the Three Kingdoms and the COVID-19-Character verified the scalability of the QA framework.

Graph theory is developing very rapidly, especially in labelling graphs. This is proven by the number of researches examining graph labelling. In graphs labelling, research focuses on developing theories. On the other hand, graph labelling (magic covering and edge super magic labelling) has a useful application. This paper will discuss the magic covering of a simple graph (Domino graph) and edge magic labelling on a simple graph (Domino graph). The conclusion is that the magic covering can be applied to a secret sharing scheme, and also edge magic labelling can be applied to ruler models.

Protasov's Theorem on the combinatorial structure of k-primitive families of non-negative matrices is generalized to k-semiprimitive matrix families. The main tool is the binary relation of colour compatibility on the vertices of the coloured graph of the matrix family. Bibliography: 14 titles.

This paper develops the theory of disconjugacy of fourth-order equations on geometric graphs which arises in modelling rod structures. The disconjugacy of an equation is defined in terms of a special fundamental system of solutions of the homogeneous equation. The disconjugacy property is shown to be related to the positivity property of the Green's functions for certain classes of boundary value problems for a fourth-order equation on a graph. A maximum principle for a fourth-order equation on a graph is formulated, and some properties of differential inequalities are proved. Bibliography: 25 titles.

In their famous 1939 paper on the physics of nuclear fission, Niels Bohr and John Wheeler presented an eye-catching graph of the fission barrier energy for various nuclides. In this paper I develop a simplified approach to developing this graph which is both sensibly true to their analysis and appropriate for a student audience.

We give a Pfaffian formula to compute the partition function of the Ising model on any graph $G$ embedded in a closed, possibly non-orientable surface. This formula, which is suitable for computational purposes, is based on the relation between the Ising model on $G$ and the dimer model on its terminal graph $G^T$. By combining the ideas of Loebl-Masbaum \cite{Loeb11}, Tesler \cite{Tes2000}, Cimasoni \cite{Cim09, Cim10} and Chelkak-Cimasoni-Kassel \cite{Chel15}, we give an elementary proof for the formula.

We will discuss the concept of a split anti-fuzzy dominating set (SAFD) in anti fuzzy graph (GAF) and investigate the relationship of γSAF (GAF)(split anti fuzzy domination number) with other known parameters of anti-fuzzy graph. Some bounds and interesting results for this parameter are obtained. The split anti-fuzzy domination on some standard anti-fuzzy graph has been discussed with some suitable graphs.

A simple connected graph is known to be Hamiltonian-t-laceable if there will be a Hamiltonian path between each pair of distinct vertices at a distance ‘t’ in G where t ∈ Z + such that 1 ≤ t ≤ diam(G). In this paper we define M-flower snark graph and discuss the hypo edge Hamiltonian laceability properties in M-flower snark graphs and Cartesian product graphs.

This article discusses the construction of a graph model of a relay system of automatic control. Based on the graph model, relations are derived for calculating transients in a relay system. To describe and analyse relay systems of automatic control, a specific feature of the relay element is used, consisting in the fact that its output value can take only certain constant values. The maximum possible set of structural states for relay elements is three. Accordingly, the relay element introduces a structure discretization effect into the system. This feature makes it possible to study relay systems with relatively simple means and makes it possible to develop methods for their calculation to a certain extent similar to methods for calculating linear systems. In developing the graph method for the description and analysis of relay automatic systems, this position was primarily taken as a basis. This method can be successfully used in any of the typical relay elements are included in the system. A two-dimensional relay system is considered as an example. Based on the graph model, the output coordinates of the system are calculated and their graphs are constructed.

A function f is called an F-centroidal mean labeling of a graph G(V, E) with p vertices and q edges if f : V(G) → {1, 2, 3, …, q + 1} is injective and the induced function g* : E(G) →{1, 2, 3, …, q} defined as g * ( u v ) = ⌊ 2 [ f ( u ) 2 + f ( u ) f ( v ) + f ( v ) 2 ] 3 [ f ( u ) + f ( v ) ] ⌋ , for all u v ∈ E ( G ) , is bijective. A graph that admits an F-centroidal mean labeling is called an F-centroidal mean graph. The line graph is one among the graph operations. In this paper, we try to analyse that the line graph operation preserves the F-centroidal mean property for the path Pn , the cycle Cn , the star graph Sn , the complete graph Kn , the graph Pn o S 1, the triangular snake graph Tn and the arbitrary subdivision of S 3.

The article deals with the organizational and technological features of construction. Methods of optimization of values to the required level are considered. Recommendations are given on making changes to organizational and technological solutions and achieving the required indicators. Reducing the timing of work while maintaining the cost of the final product is the main condition for the organization of construction production. This can be achieved by developing network and calendar schedules with the correct sequence of work, the composition of teams, and optimal technological solutions. Adjustment of graphs is possible when determining the parameters of resource graphs and comparing them with normal values. The effectiveness of the applied optimization methods in terms of time and resources depends on the technological and organizational parameters of the designed building or structure. The calculation of the parameters of construction production is carried out on the basis of the conditions that each work is provided with all the necessary resources. In reality, all resources are always limited. All planning work comes down to a constant analysis of the use of resources and their redistribution. When scheduling work, it is necessary to link the objectives of the plan in as much detail as possible with the possibilities of their implementation, that is, with available resources. Adjustment of graphs is called work to improve certain parameters of the graph. Since as a result of changing the schedules, the calculated parameters are brought into line with the task, and some optimal option is not achieved.

The researches on topological indices are initially related to graphs obtained from biological activities or chemical structures and reactivity. Recently, the research on this topic has evolved on graphs in general and even on graphs obtained from algebraic structures, such as groups, rings or modules. This paper will present various topological index concepts, various graph concepts obtained from a commutative ring and some previous studies that are relevant to those two concepts. Based on the various concepts presented, research topics related to topological indices of a graph associated with a commutative ring can be found and carried out.

Let G(V, E) be a graph with a non-empty set of vertices V and a set of edges E. A total k-labeling f: V ∪ E → {1,2, …, k} is called an edge irregular total k-labeling if the weight of each edge is distinct, where the weight of an edge e = xy is wt(e) = f(xy) + f(x) + f(y). Whereas, f is a vertex irregular total k-labeling if the weight wt(x) ≠ wt(y) for two distinct vertices x, y of V(G) where the weight wt(x) = f(x) + ∑ υx∈E(G) f(υx). The minimum k for which G has an edge irregular total k-labeling is called the total edge irregularity strength (tes) of G. Further, the total vertex irregularity strength (tvs) of G is the minimum k for which G has a vertex irregular total k-labeling. In this paper, we find tes and tvs of heptagon cactus chain graph with pendants and get the results as follows: tes ( C ( P t r 3 ) ) = ⌈ 8 r + 2 3 ⌉ and tvs ( C ( P t r 3 ) ) = ⌈ 6 r + 3 4 ⌉ .

Let F, G and H be graphs. Notation F → (G, H) means that there is any two-coloring, say red and blue, of all edges of F which contains red subgraph isomorphic to G or blue subgraph isomorphic to H. The graph F is Ramsey (G, H)-minimal if F → (G, H) but F − e ↛ (G, H) for any e ∈ E(F).The class of all Ramsey (G, H)-minimal graphs will be denoted by R(G, H). According to the previous study, we know that graphs in R(P 3, C 7) have at least 13 edges.In this paper, we find graphs in R(P 3, C 7) having seven vertices and 13 and 14 edges, respectively. Further, we give some necessary conditions for Ramsey (P 3, C 7)-minimal graphs with seven vertices.

Let G be a simple and finite graph with vertex set V and edge set E. The local antimagic total labeling of G is a map from V ∪ E to the set of positive integers from 1 to |V| + |E| such that the weight of two adjacent vertices are distinct. The weight of a vertex v is calculated by the sum of the label of the vertex v and the labels of all edges incident to it. If the vertices of G is labelled by the smallest labels, that is, {1, 2,…, |V|}, then the such labeling is called local super antimagic total labeling. Thus, the local super antimagic total labeling induces a proper vertex coloring of G where the vertex v is colored by the weight of vertex v. The minimum number of colors taken over all colorings induced by super local antimagic total labeling of G, is called local super antimagic total chromatic number of graph G, and denoted by χlsat (G). In this paper, we consider the local super antimagic total chromatic number of some wheel related graphs such as fans, even gear graphs, and sun flower graphs.

In this paper, we introduce a new notion of graph theory study, namely a local edge metric dimension. It is a natural extension of metric dimension concept. dG (e,v) = min{d(x,v),d(y,v)} is the distance between the vertex v and the edge xy in graph G. A non empty set S ⊂ V is an edge metric generator for G if for any two edges e 1 , e 2 ∈ E there is a vertex k ∈ S such that d G ( k , e 1 ≠ d G ( k , e 2 ) ) . The minimum cardinality of edge metric generator for G is called as edge metric dimension of G, denoted by dimE (G). The local edge metric dimension of G, denoted by dimE (G), is a local edge metric generator of G if r ( x k | S ) ≠ r ( y k | S ) for every pair xk,ky of adjacent edges of G. Our concern in this paper is investigating some results of local edge metric dimension on some graphs.

Let c : V(G) → ℕ be a coloring of the vertices in a graph G. For a vertex u in G, the color sum of u, denoted by σ(u), is the sum of the colors of the neighbors of u. The coloring c is called a sigma coloring of G if σ(u) ≠ σ(v) whenever u and v are adjacent vertices in G. The minimum number of colors that can be used in a sigma coloring of G is called the sigma chromatic number of G and is denoted by σ(G). Given two simple, connected graphs G and H, the corona of G and H, denoted by G ⊙ H, is the graph obtained by taking one copy of G and |V(G)| copies of H and where the ith vertex of G is adjacent to every vertex of the ith copy of H. In this study, we will show that for a graph G with |V(G)| ≥ 2, and a complete graph Kn of order n, n ≤ σ(G ⊙ Kn ) ≤ max {σ(G), n}. In addition, let Pn and Cn denote a path and a cycle of order n respectively. If m, n ≥ 3, we will prove that σ(Km ⊙ Pn ) = 2 if and only if m ≤ n − 2 ⌊ n 4 ⌋ + 2 . If n is even, we show that σ(Km ⊙ Cn ) = 2 if and only if m ≤ n − 2 ⌈ n 4 ⌉ + 2 . Furthermore, in the case that n is odd, we show that σ(Km ⊙ Cn ) = 3 if and only if m ≤ H ( ⌈ n 4 ⌉ − 1 , n − ⌈ n 4 ⌉ ) where H(r, s) denotes the number of lattice points in the convex hull of points on the plane determined by the integer parameters r and s.

A graph G is an ordered pair of sets G(V, E), where V is a set of vertices and the elements of set E are usually called edges. The concept of graph labeling has recently gained a lot of popularity in the area of graph theory. his popularity is due not only to the mathematical challenges of graph labelings but also to the wide range of applications that graph labeling offer to other brances of science, for instance, x-ray, cryptography, coding theory, circuit design and communication network design. A graph labeling we mean an assignment of integers to elements of a graph such as vertex, edge, and both. Local antimagic of a graph was motivated by Arumugam et al.. Thus, we initiate to developed the concept of local antimagic vertex coloring for subdevided graphs. A graph G called sub-devided if the graph G be the graph obtained by inserting a vertex to each edge of the graph G. Definition the concept local antimagic vertex coloring is f : E(G) → {1, 2, 3…, |E(G)|} if for any two adjacent vertices a 1 and a 2, w(a 1) = w(a 2), where for v ∈ G, w ( a ) = ∑ e ∈ E ( a ) f ( e ) , where E(a) and V(a) are respectively the set of edges incident to a. The local antimagic vertex labeling induces a proper vertex coloring of graph G if each vertex a is assigned the color w(a). The minimum colors needed to coloring the vertices in graph G called local antimagic vertex chromatic number, denoted by χla (G). In this paper we study the local antimagic vertex coloring of sub-devided some special graph as follows SFn,m , SSn,m , SWn,m and SFn,m .

Jakarta is a metropolitan city in Indonesia with high population. Geologically, its location has the potential to be affected by earthquakes from subduction zones as well as from local faults such as the Cimandiri, Lembang, and Baribis faults. Earthquake history showed that in 1780 and 1834 Jakarta was affected by an earthquake with magnitude Mw 8, destroying existing infrastructure. Several geoscientists believed the source of the fault is located in the south of Jakarta. Study with primary gravity data using the MS-SVD supported by the MS-HDVD method is a good method for identifying faults. From looking at the zero value shift on the MS-SVD graph, a fault zone is visible in the south of Jakarta. It is also confirmed by the CBA map that has contrast gravity between south and north of Jakarta. The faults have parallel-normal and parallel reverse type in East-West direction with dip angle for more than 79°. According to the fault that has been identified, south of Jakarta is evidently crossed by the fault, so the city safety needs to be considered especially for the potential of upcoming natural disasters. However, 2D forward modeling shows that subsurface Jakarta does not have density contrast between rock layers significantly.

If G, H and F are finite and simple graphs, notation F → (G, H) means that for any red-blue coloring of the edges of F, there is either a red subgraph isomorphic to G or a blue subgraph isomorphic to H. A graph F is a Ramsey (G, H)-minimal graph if F → (G, H) and for every e ∈ E(F), graph F − e ↛ (G, H). The class of all Ramsey (G, H)-minimal graphs (up to isomorphism) will be denoted by R(G, H). The characterization of all graphs in the infinite class R(P 3, Pn ) is still open, for any n ≥ 4. In this paper, we find an infinite families of trees in R(P 3, P 5). We determine how to construct unicyclic graphs in R(P 3, Pn ), for any n ≥ 5 from trees in the same class. Further, we give some properties for the unicyclic graphs constructed from trees in R(P 3, Pn ), for any n ≥ 5.

Given a graph G(V, E) with a non-empty set of vertices V and a setof edges E. A total labelling f: V ∪ E → {1,2,…, k} is called an edge irregular total labeling if the weight of every edge is distinct. The weight of anedgee, under the total labeling f, is the sum of label of edgee and all labels of vertices that are incident to e. In other words, wt(xy) = f(xy) + f(x) + f(y). The total edge irregularity strength of G, denoted by tes(G) is the minimum k used to label graph G with the edge irregular total labeling. A tadpole chain graph of length r, denoted as Tr (6, n), is a chain graph that consists of tadpole graph T (6, n) on each block. In this paper, we get t e s ( T r ( 6 , n ) ) = ⌈ ( 6 + n ) r + 2 3 ⌉ and construct an algorithm to find it.

An injective functionffrom set of vertices in graphGto a set of {0,1,…,|E| − 1} is called an odd harmonious labeling if the functionfinduced the edge functionf* from the set of edges ofGto a set of odd positive integer number {1,3,5,…,2|E| − 1} withf*(xy) =f(x) +f(y) for every edgexyinE.Graph that has an odd harmonious labeling is called odd harmonious graph. The squid graphTn,kis a graph which is obtained from a cycleCnand we addkpendant to one vertex of the cycle. It is known thatCnis an odd harmonious graph if and only ifn= 0 mod 4. However, by adding at least one pendant in the cycle graph, we can label the new graph odd harmoniously for all even number of vertices. In this paper, we showed that the graphTn,kandT2n,kare an odd harmonious graph, forn= 0 (mod 2),n≥ 4 andk≥ 1. The construction of the odd harmonious labeling of the graphTn,kandT2n,kare inspired by the odd harmonious labeling ofCnforn= 0(mod 4).

If given a graph G(V,E) with n vertices and m edges many graphs can be constructed. The graphs constructed maybe connected graphs (there exists at least one path connecting every pair of vertices in the graph) or disconnected; either simple a (contains loop or parallel edges) or not simple. In, this paper we will discuss the formula for counting the number of connected vertex labelled graph of order five (n=5) without loops, witof h minimum five edges and maa y contaiof n maximum ten parallel edges.

This paper discusses the first fundamental group from the identity graph of the group derived from the Kronecker product of the group quaternion representation. This group is called the Kronecker quaternion group. The purpose of this paper is to show the elements of the first fundamental group of the Kronecker quaternion group. A graph is needed to determine the first fundamental group. In this case, an identity graph is used; that is, graphs obtained from the Kronecker quaternion group. From this identity graph, it’s computed how many directed graphs can be made. The elements of the first fundamental group are the equivalence classes of paths in the directed identity graph that begin and end by a particular vertex. It is shown that the number of equivalence classes of paths for any fundamental group obtained from the identity graph is one plus the number of a completed graph K 3 in the identity graph of the Kronecker quaternion group.