Sin-Min Lee

Edge-magic indices of (n, n – 1)-graphs

Abstract A graph G = (V, E) with p vertices and q edges is called edge-magic if there is a bijection f : E → {1, 2, …, q} such that the induced mapping f + : V → Z p is a constant mapping, where f + (u) ≡ ∑ uv ∈ E f(uv) (mod p) . A necessary condition of edge-magicness is p ∣ q(q+1). The edge magic index of a graph G is the least positive integer k such that the k-fold of G is edge-magic. In this paper, we prove that for any multigraph G with n vertices, n − 1 edges having no loops and no isolated vertices, the k-fold of G is edge-magic if n and k satisfy a necessary condition for edge-magicness and n is odd. For n even we also have some results on full m-ary trees and spider graphs. Some counterexamples of the edge-magic indices of trees conjecture are given.

On the k-edge magic graphs

Abstract Let G be a graph with vertex set V and edge set E such that | V | = p and | E | = q . We denote this graph by (p, q)-graph. For integers k ⩾ 0 , define a one-to-one map f from E to { k , k + 1 , … , k + q − 1 } and define the vertex sum for a vertex v as the sum of the labels of the edges incident to v. If such an edge labeling induces a vertex labeling in which every vertex has a constant vertex sum (mod p), then G is said to be k-edge magic (k-EM). In this paper, we consider some specific graphs and obtain some results for them to be k-EM.

On friendly index and product-cordial index sets of Möbius-liked graph

Abstract Let G be a simple graph with vertex set V(G) and edge set E(G). Let ⟨ℤ2, +,*⟩ be a field with two elements. A vertex labeling f : V(G) → ℤ2 induces two edge labelings f+: E(G) → ℤ2 such that f+ (xy) = f(x) + f(y), whereas f* : E(G) → ℤ2 such that f* (xy) = f(x) f(y), for each edge xy ∈ E(G). For i ϵ ℤ2, let and . A labeling f of a graph G is said to be friendly if |υf (0) −υf (1)|≤ 1. The friendly index set of the graph G, denoted FI(G), is defined as the vertex labeling f is friendly}. This is a generalization of graph cordiality. The corresponding multiplicative version is called the product-cordial index set, denoted PCI(G), defined as the vertex labeling f is friendly}. In this paper, we investigate the friendly index and product-cordial index sets of a family of cubic graphs known as Möbius-liked graph, MG(n) for even n ≥ 4.

On full friendly index sets of 1-level and 2-levels N-grids

Let G be a graph with vertex set V ( G ) and edge set E ( G ) . A labeling f : V ( G ) ź Z 2 induces an edge labeling f ź : E ( G ) ź Z 2 defined by f ź ( x y ) = f ( x ) + f ( y ) , for each edge x y ź E ( G ) . For i ź Z 2 , let v f ( i ) = | { v ź V ( G ) : f ( v ) = i } | and e f ź ( i ) = | { e ź E ( G ) : f ź ( e ) = i } | . A labeling f of a graph G is said to be friendly if | v f ( 1 ) - v f ( 0 ) | ź 1 . The full friendly index set of a graph G , denoted F F I ( G ) , is defined as { e f ź ( 1 ) - e f ź ( 0 ) : the vertex labeling f is friendly } . We investigate the full friendly index sets of 1-level and 2-levels N-grids.

On friendly index sets and product-cordial index sets of gear graphs

Let G = (V,E) be a simple connected graph. A vertex labeling of f:V→{0,1} of G induces two edge labelings f+, f*:E→{0,1} defined by f+(xy) = f(x)+f(y)(mod2) and f*(xy) = f(x)f(y) for each edge xy ∈ E. For i∈{0,1}, let vf(i) = |{v∈V:f(v) = i}|, ef+(i) = |{e∈E:f+(e) = i}| and ef*(i) = |e∈E:f*(e) = i}|. A labeling f is called friendly if |vf(1)−vf(0)|≤1. The friendly index set and the product-cordial index set of G are defined as the sets {|ef+(0)−ef+(1)|:f is friendly} and {|ef*(0)−ef*(1)|:f is friendly}. In this paper, we completely determine the friendly index sets and product-cordial index sets of gear graphs. We also show that the product-cordial indices of a graph can be obtained from its adjacency matrix.