Search

Definition of graph is set pair (V(G), E(G)) where V(G) is vertex set and E(G) is edge set. A maping I: V(G) → {1,2,..., k} as label function and weight function w: V(G) → N is desined as w(u) = Σv∈N(u) l(v). The function w is called local irregularity vertex coloring if: (i) opt(l) = min(maks (li); li is label function dan (ii) for every uv ∈ E(G), w(u) ≠ w(v). The chromatic number of local irregularity vertex coloring denoted by χlis (G) is defined as χlis (G) = min{|w(V(G); w is local irregularity vertex coloring}. The method used in this paper is pattern recognition and axiomatic deductive method. In this paper, we learn local irregularity vertex coloring of vertex amalgamation of path graph and determine the chromatic number on local irregularity vertex coloring of vertex amalgamation of path graph. This paper use vertex amalgamation of path graph (amal(Pn, v, m)). The result of this study are expected to be used as basic studies and science development as well as applications related to local irregularity vertex coloring of vertex amalgamation of path graph. [ABSTRACT FROM AUTHOR]

Let G be a simple graph and connected. A function l : V (G) → { 1 , 2 , ... , k } is called vertex irregular k -labeling and w : V (G) → N where w (u) = ∑ v ∈ N (u) l (v). The function of f is called local irregular vertex coloring if every u v ∈ E (G) , w (u) ≠ w (v) and opt (l) = min { max l i ; l i vertex irregular labeling}. The local irregular chromatic number is denoted by χ lis (G). In this paper, we study local irregular vertex coloring of S m ▹ v 0 S n , S m ▹ v 1 S n , and S m ▹ v 1 P n. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a simple, finite, undirected, and connected graph with vertex set V (G) and edge set E (G). A bijection l : V (G) → { 1 , 2 , ... , k } is label function l if opt (l) = min { max (l i) : l i   v e r t e x   i r r e g u l a r   l a b e l i n g } and for any two adjacent vertices u and v , w (u) ≠ w (v) where w (u) = ∑ v ∈ N (u) l (v) and N (u) is set of vertices adjacent to v. w is called local irregularity vertex coloring. The minimum cardinality of local irregularity vertex coloring of G is called chromatic number local irregular denoted by χ l i s (G). In this paper, we verify the exact values of volcano, broom, parachute, double broom and complete multipartite graphs. [ABSTRACT FROM AUTHOR]

All graphs in this paper are undirected and connected graphs. An ordered k-partition set Π = { S 1 , S 2 , ... , S k } , where S i ⊂ V (G) , 1 ≤ i ≤ k , the representation of a vertex v of G with respect to Π is r (v | Π) = (d (v , S 1) , d (v , S 2) , ... , d (v , S k)) , where d (v , S i) is the distance between the vertex v and the set S i with d (v , S i) = min { d (v , x) ; x ∈ S i } for 1 ≤ i ≤ k. The partition set Π is a local resolving partition of G if r (v , Π) ≠ r (u | Π) for u adjacent to v of G. The minimum local resolving partition Π is a local partition dimension of G , denoted by lpd (G). In our paper, we found the sharp bounds of the local partition dimension of G and determine the exact value of some special graph. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a graph. A proper k -coloring of graph G is r -dynamic coloring if for every v , the neighbors of vertex v receive at least min { r , d (v) } different colors. The minimum k such that graph G has r -dynamic k coloring is called the r -dynamic chromatic number, denoted by χ r (G). In this paper, we study the r -dynamic coloring of corona product of graph. The corona product of graph is obtained by taking a number of vertices | V (G) | copy of H , and making the i th of V (G) adjacent to every vertex of the i th copy of V (H). We obtain the lower bound of r -dynamic chromatic number of corona product of graphs and some exact value. [ABSTRACT FROM AUTHOR]

For a set W = { s 1 , s 2 , ... , s k } of vertices of a graph G , the representation multiset of a vertex v of G with respect to W is r (v | W) = { d (v , s 1) , d (v , s 2) , ... , d (v , s k) } , where d (v , s i) is a distance between of the vertex v and the vertices in W together with their multiplicities. The set W is a resolving set of G if r (v | W) ≠ r (u | W) for every pair u , v of distinct vertices of G. The minimum resolving set W is a multiset basis of G. If G has a multiset basis, then its cardinality is called multiset dimension, denoted by md (G). A set W of vertices in G is a dominating set for G if every vertex of G that is not in W is adjacent to some vertex of W. The minimum cardinality of the dominating set is a domination number, denoted by γ (G). A vertex set of some vertices in G that is both resolving and dominating set is a resolving dominating set. The minimum cardinality of resolving dominating set is called resolving domination number, denoted by γ r (G). In our paper, we investigate and establish sharp bounds of the resolving domination number of G and determine the exact value of some family graphs. [ABSTRACT FROM AUTHOR]

We consider V, E are respectively vertex and edge sets of a simple, nontrivial and connected graph G. For an ordered set W = {w1, w2, w3, ..., wk} of vertices and a vertex v ∈ G, the ordered r(v|W) = (d(v, w1), d(v, w2), ..., d(v, wk)) of k-vector is representations of v with respect to W, where d(v, w) is the distance between the vertices v and w. The set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W. The metric dimension, denoted by dim(G) is min of |W|. Furthermore, the resolving set W of graph G is called non-isolated resolving set if there is no ∀v ∈ W induced by non-isolated vertex. While a non-isolated resolving number, denoted by nr(G), is the minimum cardinality of non-isolated resolving set in graph. In this paper, we study the non isolated resolving number of graph with any pendant edges. [ABSTRACT FROM AUTHOR]

One of the topics of distance in graphs is resolving set problem. This topic has many applications in science and technology namely navigation robots, chemistry structure, and computer sciences. Suppose the set W = {s1,s2,…,sk}⊂ V (G), the vertex representations of x ∈ V (G) is rm(x|W) = {d(x,s1),d(x,s2),…,d(x,sk)}, where d(x,si) is the length of the shortest path of the vertex x and the vertex in W together with their multiplicity. The set W is called a local m-resolving set of graphs G if rm(v|W)≠rm(u|W) for uv ∈ E(G). The local m-resolving set having minimum cardinality is called the local multiset basis and its cardinality is called the local multiset dimension of G, denoted by mdl(G). In our paper, we determine the establish bounds of local multiset dimension of graph resulting corona product of tree graphs. [ABSTRACT FROM AUTHOR]