In the related literature, the eccentricities of graphs have been studied recently. The main purp... more In the related literature, the eccentricities of graphs have been studied recently. The main purpose of this paper is to discuss the eccentric spectrum of a graph. For any two vertices u and v in a connected graph G, d G (u,v) denotes the distance between vertices u and v. The eccentricity e G (v) of a vertex v in G is the maximum number of d G (v,u) over all vertex u. A vertex u is an eccentric vertex if there exists a vertex v such that e G (v)=d G (v,u). A number k is called an eccentric number of G if, for each vertex v with e G (v)=k, v is an eccentric vertex. The eccentric spectrum S G of a connected graph G is a set of all eccentric numbers in G. If d is the diameter of G, then d∈S G . In the paper, we show that for positive integers r≤d≤2r and d∈S⊆{r,r+1,⋯,d}, there exists a connected graph G with radius r, diameter d and eccentric spectrum S. This result also proves the conjecture of G. Chartrand, M. Schultz and S. J. Winters in [Networks 28, No.4, 181-186 (1996; Zbl 0873.9...
The linear 2-arboricity of a graph is the smallest k such that the graph can be partitioned into ... more The linear 2-arboricity of a graph is the smallest k such that the graph can be partitioned into k edge-disjoint forests the components of which are paths of length at most 2. The authors prove that the linear 2-arboricity of an outerplanar graph of maximum degree D is at most ⌊(D+4)/2⌋.
A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram of a finit... more A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram of a finite partially ordered set. The direct product G X H of graphs G and H is the graph having vertex set V(G) X V(H) and edge set E(G X H) = {(g_i,h_s)(g_j,h_t): g_ig_j belongs to E(G) and h_sh_t belongs to E(H)}. We prove that the direct product M_m(G) X M_n(H) of the generalized Mycielskians of G and H is a cover graph if and only if G or H is bipartite.
IEEE Conference on Industrial Electronics and Applications, 2010
Myopia is a growing concern in many societies. In extremely high myopia, pathological myopia, whi... more Myopia is a growing concern in many societies. In extremely high myopia, pathological myopia, which can cause visual loss, can occur. Pathological myopia is also accompanied by various visually perceivable symptoms on the retina, such as peripapillary atrophy. PAMELA is an automatic system for the detection of pathological myopia through the presence of peripapillary atrophy. In this paper, we describe
Ethylene vinyl acetate (EVA) copolymer was emulsified in the melt state using hydrophobically mod... more Ethylene vinyl acetate (EVA) copolymer was emulsified in the melt state using hydrophobically modified water soluble polymers (HMWSPs) as surfactants and the flow induced phase inversion (FIPI) emulsification technique. The history of the emulsification and emulsion structure were monitored using a process rheometer and off-line scanning electron microscopy and particle size measurements. It was shown that low molecular weight surfactants
In recent years, many multistage interconnection networks using 2 1 2 switching elements have bee... more In recent years, many multistage interconnection networks using 2 1 2 switching elements have been proposed for parallel architectures. Typical examples are baseline networks, banyan networks, shuffle-exchange networks, and their inverses. As these networks are blocking, such networks with extra stages have also been studied extensively. These include Benes networks and D ! D* networks. Re- cently, Hwang et al. studied k-extra-stage networks, which are a generalization of the above networks. They also investigated the equivalence issue among some of these networks. In this paper, we studied a more general class of networks, which we call (m / 1)-stage d-nary bit permutation networks. We characterize the equivalence of such networks by sequence of positive integers. q 1999 John Wiley & Sons, Inc. Networks 33: 261-267, 1999
ABSTRACT In the paper, we study the hamiltonian numbers in digraphs. A hamiltonian walk of a digr... more ABSTRACT In the paper, we study the hamiltonian numbers in digraphs. A hamiltonian walk of a digraph D is a closed spanning directed walk with minimum length in D. The length of a hamiltonian walk of a digraph D is called the hamiltonian number of D, denoted h(D). We prove that if a digraph D of order n is strongly connected, then $n\leq h(D)\leq\lfloor\frac{(n+1)^{2}}{4} \rfloor$ , and hence characterize the strongly connected digraphs of order n with hamiltonian number $\lfloor\frac{(n+1)^{2}}{4} \rfloor$ . In addition, we show that for each k with $4\leq n\leq k\leq\lfloor \frac{(n+1)^{2}}{4} \rfloor$ , there exists a digraph with order n and hamiltonian number k. Furthermore, we also study the hamiltonian spectra of graphs.
ABSTRACT For the study of hamiltonicity of graphs and digraphs, Goodman and Hedetniemi introduced... more ABSTRACT For the study of hamiltonicity of graphs and digraphs, Goodman and Hedetniemi introduced the concept of Hamiltonian number. The Hamiltonian number h(D) of a digraph D is the minimum length of a closed walk containing all vertices of D. In this paper, we study Hamiltonian numbers of the following proposed networks, which include strongly connected double loop networks. For integers d≥1, m≥1 and ℓ≥0, the Möbius double loop network MDL(d,m,ℓ) is the digraph with vertex set {(i,j):0≤i≤d−1,0≤j≤m−1} and arc set {(i,j)(i+1,j) or (i,j)(i+1,j+1):0≤i≤d−2,0≤j≤m−1}∪{(d−1,j)(0,j+ℓ) or (d−1,j)(0,j+ℓ+1):0≤j≤m−1}, where the second coordinate y of a vertex (x,y) is taken modulo m. We give an upper bound for the Hamiltonian number of a Möbius double loop network. We also give a necessary and sufficient condition for a Möbius double loop network MDL(d,m,ℓ) to have Hamiltonian number at most dm, dm+d, dm+1 or dm+2.
ABSTRACT For each vertex upsilon in a graph G, the maximum length of a cycle which passes through... more ABSTRACT For each vertex upsilon in a graph G, the maximum length of a cycle which passes through D is called the cycle number of upsilon, denoted by c(upsilon). A sequence a(1),a(2),...,a(n) of nonnegative integers is called a cycle sequence of a graph G if the vertices of G can be labeled as upsilon (1),upsilon (2),...,upsilon (n) such that a(i) = c(upsilon (i)) for 1 less than or equal to i less than or equal to n. We give;some sufficient and necessary conditions for a sequence to be a cycle sequence. We can thereby derive a polynomial time procedure for recognizing cycle sequences.
ABSTRACT Suppose D is an acyclic orientation of a graph G. An arc of D is said to be independent ... more ABSTRACT Suppose D is an acyclic orientation of a graph G. An arc of D is said to be independent if its reversal results another acyclic orientation. Denote i(D) the number of independent arcs in D and N(G) = {i(D) : D is an acyclic orientation of G}. Also, let imin(G) be the minimum of N(G) and imax(G) the maximum. While it is known that imin(G) = |V (G)| 1 for any connected graph G, the present paper determines imax(G) for complete r-partite graphs G. We then determine N(G) for any balanced complete r-partite graph G, showing that N(G) is not a set of consecutive integers. This answers a question of West's. Finally, we give some complete r-partite graphs G whose N(G) is a set of consecutive integers.
ABSTRACT For an oriented graph D, let ID[u,v] denote the set of all vertices lying on a u–v geode... more ABSTRACT For an oriented graph D, let ID[u,v] denote the set of all vertices lying on a u–v geodesic or a v–u geodesic. For S⊆V(D), let ID[S] denote the union of all ID[u,v] for all u,v∈S. Let [S]D denote the smallest convex set containing S. The geodetic number g(D) of an oriented graph D is the minimum cardinality of a set S with ID[S]=V(D) and the hull number h(D) of an oriented graph D is the minimum cardinality of a set S with [S]D=V(D). For a connected graph G, let O(G) be the set of all orientations of G, define g−(G)=min{g(D):D∈O(G)}, g+(G)=max{g(D):D∈O(G)}, h−(G)=min{h(D):D∈O(G)}, and h+(G)=max{h(D):D∈O(G)}. By the above definitions, h−(G)≤g−(G) and h+(G)≤g+(G). In the paper, we prove that g−(G)h+(G) for a connected graph G of order at least 3, and for any nonnegative integers a and b, there exists a connected graph G such that g−(G)−h−(G)=a and g+(G)−h+(G)=b. These results answer a problem of Farrugia in [A. Farrugia, Orientable convexity, geodetic and hull numbers in graphs, Discrete Appl. Math. 148 (2005) 256–262].
ABSTRACT A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram o... more ABSTRACT A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram of a finite partially ordered set. We prove that the generalized Mycielski graphs Mm(C2t+1) of an odd cycle, Kneser graphs KG(n,k), and Schrijver graphs SG(n,k) are not cover graphs when m⩾0,t⩾1, k⩾1, and n⩾2k+2. These results have consequences in circular chromatic number.
ABSTRACT The well-known theorem by Gallai-Roy-Vitaver says that every digraph G has a directed pa... more ABSTRACT The well-known theorem by Gallai-Roy-Vitaver says that every digraph G has a directed path with at least χ(G) vertices; hence this holds also for graphs. Li strengthened the digraph result by showing that the directed path can be constrained to start from any vertex that can reach all others. For a graph G given a proper χ(G)-coloring, he proved that the path can be required to start at any vertex and visit vertices of all colors. We give a shorter proof of this. He conjectured that the same holds for digraphs; we provide a strongly connected counterexample. We also give another extension of the Gallai-Roy-Vitaver Theorem on graphs.
In the related literature, the eccentricities of graphs have been studied recently. The main purp... more In the related literature, the eccentricities of graphs have been studied recently. The main purpose of this paper is to discuss the eccentric spectrum of a graph. For any two vertices u and v in a connected graph G, d G (u,v) denotes the distance between vertices u and v. The eccentricity e G (v) of a vertex v in G is the maximum number of d G (v,u) over all vertex u. A vertex u is an eccentric vertex if there exists a vertex v such that e G (v)=d G (v,u). A number k is called an eccentric number of G if, for each vertex v with e G (v)=k, v is an eccentric vertex. The eccentric spectrum S G of a connected graph G is a set of all eccentric numbers in G. If d is the diameter of G, then d∈S G . In the paper, we show that for positive integers r≤d≤2r and d∈S⊆{r,r+1,⋯,d}, there exists a connected graph G with radius r, diameter d and eccentric spectrum S. This result also proves the conjecture of G. Chartrand, M. Schultz and S. J. Winters in [Networks 28, No.4, 181-186 (1996; Zbl 0873.9...
The linear 2-arboricity of a graph is the smallest k such that the graph can be partitioned into ... more The linear 2-arboricity of a graph is the smallest k such that the graph can be partitioned into k edge-disjoint forests the components of which are paths of length at most 2. The authors prove that the linear 2-arboricity of an outerplanar graph of maximum degree D is at most ⌊(D+4)/2⌋.
A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram of a finit... more A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram of a finite partially ordered set. The direct product G X H of graphs G and H is the graph having vertex set V(G) X V(H) and edge set E(G X H) = {(g_i,h_s)(g_j,h_t): g_ig_j belongs to E(G) and h_sh_t belongs to E(H)}. We prove that the direct product M_m(G) X M_n(H) of the generalized Mycielskians of G and H is a cover graph if and only if G or H is bipartite.
IEEE Conference on Industrial Electronics and Applications, 2010
Myopia is a growing concern in many societies. In extremely high myopia, pathological myopia, whi... more Myopia is a growing concern in many societies. In extremely high myopia, pathological myopia, which can cause visual loss, can occur. Pathological myopia is also accompanied by various visually perceivable symptoms on the retina, such as peripapillary atrophy. PAMELA is an automatic system for the detection of pathological myopia through the presence of peripapillary atrophy. In this paper, we describe
Ethylene vinyl acetate (EVA) copolymer was emulsified in the melt state using hydrophobically mod... more Ethylene vinyl acetate (EVA) copolymer was emulsified in the melt state using hydrophobically modified water soluble polymers (HMWSPs) as surfactants and the flow induced phase inversion (FIPI) emulsification technique. The history of the emulsification and emulsion structure were monitored using a process rheometer and off-line scanning electron microscopy and particle size measurements. It was shown that low molecular weight surfactants
In recent years, many multistage interconnection networks using 2 1 2 switching elements have bee... more In recent years, many multistage interconnection networks using 2 1 2 switching elements have been proposed for parallel architectures. Typical examples are baseline networks, banyan networks, shuffle-exchange networks, and their inverses. As these networks are blocking, such networks with extra stages have also been studied extensively. These include Benes networks and D ! D* networks. Re- cently, Hwang et al. studied k-extra-stage networks, which are a generalization of the above networks. They also investigated the equivalence issue among some of these networks. In this paper, we studied a more general class of networks, which we call (m / 1)-stage d-nary bit permutation networks. We characterize the equivalence of such networks by sequence of positive integers. q 1999 John Wiley & Sons, Inc. Networks 33: 261-267, 1999
ABSTRACT In the paper, we study the hamiltonian numbers in digraphs. A hamiltonian walk of a digr... more ABSTRACT In the paper, we study the hamiltonian numbers in digraphs. A hamiltonian walk of a digraph D is a closed spanning directed walk with minimum length in D. The length of a hamiltonian walk of a digraph D is called the hamiltonian number of D, denoted h(D). We prove that if a digraph D of order n is strongly connected, then $n\leq h(D)\leq\lfloor\frac{(n+1)^{2}}{4} \rfloor$ , and hence characterize the strongly connected digraphs of order n with hamiltonian number $\lfloor\frac{(n+1)^{2}}{4} \rfloor$ . In addition, we show that for each k with $4\leq n\leq k\leq\lfloor \frac{(n+1)^{2}}{4} \rfloor$ , there exists a digraph with order n and hamiltonian number k. Furthermore, we also study the hamiltonian spectra of graphs.
ABSTRACT For the study of hamiltonicity of graphs and digraphs, Goodman and Hedetniemi introduced... more ABSTRACT For the study of hamiltonicity of graphs and digraphs, Goodman and Hedetniemi introduced the concept of Hamiltonian number. The Hamiltonian number h(D) of a digraph D is the minimum length of a closed walk containing all vertices of D. In this paper, we study Hamiltonian numbers of the following proposed networks, which include strongly connected double loop networks. For integers d≥1, m≥1 and ℓ≥0, the Möbius double loop network MDL(d,m,ℓ) is the digraph with vertex set {(i,j):0≤i≤d−1,0≤j≤m−1} and arc set {(i,j)(i+1,j) or (i,j)(i+1,j+1):0≤i≤d−2,0≤j≤m−1}∪{(d−1,j)(0,j+ℓ) or (d−1,j)(0,j+ℓ+1):0≤j≤m−1}, where the second coordinate y of a vertex (x,y) is taken modulo m. We give an upper bound for the Hamiltonian number of a Möbius double loop network. We also give a necessary and sufficient condition for a Möbius double loop network MDL(d,m,ℓ) to have Hamiltonian number at most dm, dm+d, dm+1 or dm+2.
ABSTRACT For each vertex upsilon in a graph G, the maximum length of a cycle which passes through... more ABSTRACT For each vertex upsilon in a graph G, the maximum length of a cycle which passes through D is called the cycle number of upsilon, denoted by c(upsilon). A sequence a(1),a(2),...,a(n) of nonnegative integers is called a cycle sequence of a graph G if the vertices of G can be labeled as upsilon (1),upsilon (2),...,upsilon (n) such that a(i) = c(upsilon (i)) for 1 less than or equal to i less than or equal to n. We give;some sufficient and necessary conditions for a sequence to be a cycle sequence. We can thereby derive a polynomial time procedure for recognizing cycle sequences.
ABSTRACT Suppose D is an acyclic orientation of a graph G. An arc of D is said to be independent ... more ABSTRACT Suppose D is an acyclic orientation of a graph G. An arc of D is said to be independent if its reversal results another acyclic orientation. Denote i(D) the number of independent arcs in D and N(G) = {i(D) : D is an acyclic orientation of G}. Also, let imin(G) be the minimum of N(G) and imax(G) the maximum. While it is known that imin(G) = |V (G)| 1 for any connected graph G, the present paper determines imax(G) for complete r-partite graphs G. We then determine N(G) for any balanced complete r-partite graph G, showing that N(G) is not a set of consecutive integers. This answers a question of West's. Finally, we give some complete r-partite graphs G whose N(G) is a set of consecutive integers.
ABSTRACT For an oriented graph D, let ID[u,v] denote the set of all vertices lying on a u–v geode... more ABSTRACT For an oriented graph D, let ID[u,v] denote the set of all vertices lying on a u–v geodesic or a v–u geodesic. For S⊆V(D), let ID[S] denote the union of all ID[u,v] for all u,v∈S. Let [S]D denote the smallest convex set containing S. The geodetic number g(D) of an oriented graph D is the minimum cardinality of a set S with ID[S]=V(D) and the hull number h(D) of an oriented graph D is the minimum cardinality of a set S with [S]D=V(D). For a connected graph G, let O(G) be the set of all orientations of G, define g−(G)=min{g(D):D∈O(G)}, g+(G)=max{g(D):D∈O(G)}, h−(G)=min{h(D):D∈O(G)}, and h+(G)=max{h(D):D∈O(G)}. By the above definitions, h−(G)≤g−(G) and h+(G)≤g+(G). In the paper, we prove that g−(G)h+(G) for a connected graph G of order at least 3, and for any nonnegative integers a and b, there exists a connected graph G such that g−(G)−h−(G)=a and g+(G)−h+(G)=b. These results answer a problem of Farrugia in [A. Farrugia, Orientable convexity, geodetic and hull numbers in graphs, Discrete Appl. Math. 148 (2005) 256–262].
ABSTRACT A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram o... more ABSTRACT A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram of a finite partially ordered set. We prove that the generalized Mycielski graphs Mm(C2t+1) of an odd cycle, Kneser graphs KG(n,k), and Schrijver graphs SG(n,k) are not cover graphs when m⩾0,t⩾1, k⩾1, and n⩾2k+2. These results have consequences in circular chromatic number.
ABSTRACT The well-known theorem by Gallai-Roy-Vitaver says that every digraph G has a directed pa... more ABSTRACT The well-known theorem by Gallai-Roy-Vitaver says that every digraph G has a directed path with at least χ(G) vertices; hence this holds also for graphs. Li strengthened the digraph result by showing that the directed path can be constrained to start from any vertex that can reach all others. For a graph G given a proper χ(G)-coloring, he proved that the path can be required to start at any vertex and visit vertices of all colors. We give a shorter proof of this. He conjectured that the same holds for digraphs; we provide a strongly connected counterexample. We also give another extension of the Gallai-Roy-Vitaver Theorem on graphs.
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