Wai Shiu
Hong Kong Baptist University, Mathematics, Department Member
Let \(G = (V,E)\) be a simple graph of order \(p\) and size \(q\). A graph \(G\) is called local antimagic (total) if \(G\) admits a local antimagic (total) labeling. A bijection \(g : E \to \{1,2,\ldots,q\}\) is called a local antimagic... more
Let \(G = (V,E)\) be a simple graph of order \(p\) and size \(q\). A graph \(G\) is called local antimagic (total) if \(G\) admits a local antimagic (total) labeling. A bijection \(g : E \to \{1,2,\ldots,q\}\) is called a local antimagic labeling of \(G\) if for any two adjacent vertices \(u\) and \(v\), we have \(g^+(u) \neq g^+(v)\), where \(g^+(u) = \sum_{e\in E(u)} g(e)\), and \(E(u)\) is the set of edges incident to \(u\). Similarly, a bijection \(f:V(G)\cup E(G)\to \{1,2,\ldots,p+q\}\) is called a local antimagic total labeling of \(G\) if for any two adjacent vertices \(u\) and \(v\), we have \(w_f(u)\neq w_f(v)\), where \(w_f(u) = f(u) + \sum_{e\in E(u)} f(e)\). Thus, any local antimagic (total) labeling induces a proper vertex coloring of \(G\) if vertex \(v\) is assigned the color \(g^+(v)\) (respectively, \(w_f(u)\)). The local antimagic (total) chromatic number, denoted \(\chi_{la}(G)\) (respectively \(\chi_{lat}(G)\)), is the minimum number of induced colors taken over ...
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These are notes of a course given in Fall, 2007 and 2008 to the Honors sections of our elementary linear algebra course. Their comments and corrections have greatly improved the exposition.
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Let G be a connected (molecule) graph. The Wiener index W G and Kirchhoff index K f G of G are defined as the sum of distances and the resistance distances between all unordered pairs of vertices in G , respectively. In this paper,... more
Let G be a connected (molecule) graph. The Wiener index W G and Kirchhoff index K f G of G are defined as the sum of distances and the resistance distances between all unordered pairs of vertices in G , respectively. In this paper, explicit formulae for the expected values of the Wiener and Kirchhoff indices of random pentachains are derived by the difference equation and recursive method. Based on these formulae, we then make comparisons between the expected values of the Wiener index and the Kirchhoff index in random pentachains and present the average values of the Wiener and Kirchhoff indices with respect to the set of all random pentachains with n pentagons.
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Abstract The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. P. Hansen and D. Stevanovic (2008) [9] determined the graphs with maximum spectral radius among all connected graphs of order n with diameter D .... more
Abstract The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. P. Hansen and D. Stevanovic (2008) [9] determined the graphs with maximum spectral radius among all connected graphs of order n with diameter D . In this paper, we generalize this result to k -connected graphs of order n with diameter D .
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Incidence coloring of a graph G is a mapping from the set of incidences to a color-set C such that adjacent incidences are assigned distinct colors. In the last ten years, incidence coloring was developed independently. To link up... more
Incidence coloring of a graph G is a mapping from the set of incidences to a color-set C such that adjacent incidences are assigned distinct colors. In the last ten years, incidence coloring was developed independently. To link up incidence coloring with edge coloring, we proved that if an odd degree regular graph is (Δ+1)-incidence colorable, then it is Δ-edge colorable. This result helps to establish some sufficient conditions for the cubic graphs that are not (Δ+1)-incidence colorable. Moreover, two kinds of-cubic graphs are proved to be 4-incidence colorable.
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Let G be a connected simple (p, q)graph and ka non-negative integer. The graph G is said to be kedge-graceful if the edges can be labeled with k, k+ 1,..., k+ q− 1 so that the vertex sums are distinct modulo p. The set of all such k... more
Let G be a connected simple (p, q)graph and ka non-negative integer. The graph G is said to be kedge-graceful if the edges can be labeled with k, k+ 1,..., k+ q− 1 so that the vertex sums are distinct modulo p. The set of all such k where G is kedge-graceful is ...
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ABSTRACT Let A be a non-trivial abelian group and A * =A-{0}. A graph is A-magic if there exists an edge-labeling using elements of A * which induces a constant vertex labeling of the graph. Although a fair amount of research has been... more
ABSTRACT Let A be a non-trivial abelian group and A * =A-{0}. A graph is A-magic if there exists an edge-labeling using elements of A * which induces a constant vertex labeling of the graph. Although a fair amount of research has been done on A-magic labelings of graphs, there is much which is still unknown. In this paper, we construct a large collection of non-intuitive examples and counterexamples, which provide further insight into the integer-magic spectra of graphs. Particular attention is devoted to the integer-magic spectra of products of graphs.
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Let G be a graph of order n. Let ?1 , ?2 , . . . , ?n be the eigenvalues of the adjacency matrix of G, and let ?1 , ?2 , . . . , ?n be the eigenvalues of the Laplacian matrix of G. Much studied Estrada index of the graph G is defined n as... more
Let G be a graph of order n. Let ?1 , ?2 , . . . , ?n be the eigenvalues of the adjacency matrix of G, and let ?1 , ?2 , . . . , ?n be the eigenvalues of the Laplacian matrix of G. Much studied Estrada index of the graph G is defined n as EE = EE(G)= ?n/i=1 e?i . We define and investigate the Laplacian Estrada index of the graph G, LEE=LEE(G)= ?n/i=1 e(?i - 2m/n). Bounds for LEE are obtained, as well as some relations between LEE and graph Laplacian energy.
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The Szeged index (Sz) is a recently proposed structure descriptor based on distances between vertices of the molecular graph. Numerous properties of Sz were previously shown to parallel that of the long-known Wiener index (W). We report... more
The Szeged index (Sz) is a recently proposed structure descriptor based on distances between vertices of the molecular graph. Numerous properties of Sz were previously shown to parallel that of the long-known Wiener index (W). We report explicit combinatorial expressions for Sz of a variety of homologous series of benzenoid hydrocarbons. Comparing Sz with W it is found that in
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It is shown that some transformations change the largest Laplacian eigenvalue of a tree T.By applying these results,we get the order of trees by their largest Laplacian eigenvalues.
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ABSTRACT Let A be an abelian group. An A-magic of a graph G=(V,E) is a labeling l:E(G)→A∖{0} such that the sum of the labels of the edges incident with u∈V is a constant, where 0 is the identity element of the group A. We show that some... more
ABSTRACT Let A be an abelian group. An A-magic of a graph G=(V,E) is a labeling l:E(G)→A∖{0} such that the sum of the labels of the edges incident with u∈V is a constant, where 0 is the identity element of the group A. We show that some classes of graphs are A-magic for all abelian groups A of even order other than 2. Also, we prove that product and composition of A-magic graphs are also A-magic.
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ABSTRACT In this note, we will focus on several applications on the Dirichlet's box principle in Discrete Mathematics lesson and number theory lesson. In addition, the main result is an innovative game on a triangular board... more
ABSTRACT In this note, we will focus on several applications on the Dirichlet's box principle in Discrete Mathematics lesson and number theory lesson. In addition, the main result is an innovative game on a triangular board developed by the authors. The game has been used in teaching and learning mathematics in Discrete Mathematics and some high schools in Hong Kong. (Contains 1 note.)
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