Browsing by Author "Lin, Chiang"

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Antimagicness of star forests
(2020-03-01)
Chen, Zhen-Chun
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Huang, Kuo-Ching
;
Lin, Chiang
;
Shang, Jen-Ling
;
Lee, Ming-Ju
An edge labeling of a graph G is a bijection f : E(G) {1, 2, " " ", 1E(G)1}. The induced vertex sum f+ of f is a function defined on V(G) given by f+ (u) = E(G) f(uv) for all u E V(G). The value f+(u) is called the vertex sum at u. The graph G is called antimagic if there exists an edge labeling of G such that the vertex sums at all vertices of G are distinct. A star forest is the union of disjoint stars. Let S denote a star with n edges, and mS denote a star forest consisting of the disjoint m copies of S. It is known that mS2 is antimagic if and only if m = 1. In this study, a necessary condition and a sufficient condition are obtained whereby a star forest mS2 U Snr U Snz U " " " U Sn, (ni, n2, " " ", nk > 3) may be antimagic. In addition, a necessary and sufficient condition is obtained whereby a star forest mS2 US (n > 3) may be antimagic. Moreover, a graph consisting of disjoint stars together with an extra disjoint path is also verified to be antimagic.
Maximum variances and minimum statuses of connected Weighted Graphs
(2017-09-01)
Lin, Chiang
;
Tsai, Wei-Han
;
Shang, Jen-Ling
;
Lee, Ming-Ju
;
應用數學系
Let G be a connected graph. For a permutation of V(G), the variance v(G)(phi) due to phi is the sum of d(G)(x,phi(x)) (x is an element of V(G)). The maximum variance Mv(G) of G is the maximum of v(G)(phi) (phi is a permutation of V(G)). For a vertex x in G, the status s(G)(x) of x is the sum of d(G)(x,y) (y is an element of V(G)). The minimum status ms(G) of G is the minimum of s(G)(x) (x is an element of V(G)). A vertex x in G is said to be a vertex with 1/2-property, if |V(G')| <= 1/2|V(G)| for every component G' of G - x. A weighted graph (G, w) is a graph G with a weight function w defined on E(G). The notions of maximum variance and minimum status are extended to connected weighted graphs. Let Mv(G, w) and ms(G, w) denote the maximum variance and the minimum status, respectively, of a connected weighted graph (G, w). In section 2, we prove that if a connected weighted graph (G, w) contains a vertex with 1/2-property, then Mv(G,w) = 2ms(G,w). We also give a criterion of bipartite graphs in terms of variance, and investigate the variance spectrum of a connected graph which contains a vertex with 1/2-property. In section 3, we obtain the formulas for the maximum variance and the minimum status, respectively, of the Cartesian product of two connected weighted graphs.