For a graph G and two positive integers j and k an m-L j k -edge-labeling of G is an assignment from the set 0 1 … m-to the edges such that adjacent edges receive labels that differ by at least j and edges at distanc...For a graph G and two positive integers j and k an m-L j k -edge-labeling of G is an assignment from the set 0 1 … m-to the edges such that adjacent edges receive labels that differ by at least j and edges at distance two receive labels that differ by at least k.Theλ′j k-number of G denoted byλ′j k G is the minimum integer m overall m-L j k -edge-labeling of G.The necklace is a specific type of Halin graph.The L 1 2 -edge-labeling of necklaces is studied and the lower and upper bounds on λ′1 2-number for necklaces are given.Also both the lower and upper bounds are attainable.展开更多
Let j, k and m be three positive integers, a circular m-L（j, k）-labeling of a graph G is a mapping f： V（G）→{0, 1, …, m-1}such that f（u）-f（v）m≥j if u and v are adjacent, and f（u）-f（v）m≥k if u and v are...Let j, k and m be three positive integers, a circular m-L（j, k）-labeling of a graph G is a mapping f： V（G）→{0, 1, …, m-1}such that f（u）-f（v）m≥j if u and v are adjacent, and f（u）-f（v）m≥k if u and v are at distance two,where a-bm=min{a-b,m-a-b}. The minimum m such that there exists a circular m-L（j, k）-labeling of G is called the circular L（j, k）-labeling number of G and is denoted by σj, k（G）. For any two positive integers j and k with j≤k,the circular L（j, k）-labeling numbers of trees, the Cartesian product and the direct product of two complete graphs are determined.展开更多
L（ s, t）-labeling is a variation of graph coloring which is motivated by a special kind of the channel assignment problem. Let s and t be any two nonnegative integers. An L （s, t）-labeling of a graph G is an assig...L（ s, t）-labeling is a variation of graph coloring which is motivated by a special kind of the channel assignment problem. Let s and t be any two nonnegative integers. An L （s, t）-labeling of a graph G is an assignment of integers to the vertices of G such that adjacent vertices receive integers which differ by at least s, and vertices that are at distance of two receive integers which differ by at least t. Given an L（s, t） -labeling f of a graph G, the L（s, t） edge span of f, βst （ G, f） = max { ｜f（u） -f（v）｜： （ u, v） ∈ E（G） } is defined. The L（ s, t） edge span of G, βst（G）, is minβst（G,f）, where the minimum runs over all L（s, t）-labelings f of G. Let T be any tree with a maximum degree of △≥2. It is proved that if 2s≥t≥0, then βst（T） =（ [△/2 ] - 1）t ＋s; if 0≤2s 〈 t and △ is even, then βst（T） = [ （△ - 1） t/2 ] ; and if 0 ≤2s 〈 t and △ is odd, then βst（T） = （△ - 1） t/2 ＋ s. Thus, the L（s, t） edge spans of the Cartesian product of two paths and of the square lattice are completely determined.展开更多
Multi-component signals contain multiple signal parts expressed in the same physical modality. One way to identify individual components is if they are produced by different physical mechanisms. Here, I studied the me...Multi-component signals contain multiple signal parts expressed in the same physical modality. One way to identify individual components is if they are produced by different physical mechanisms. Here, I studied the mechanisms generating acoustic signals in the courtship displays of the Calliope hummingbird Stellula calliope. Display dives consisted of three synchronized sound elements, a high-frequency tone （hfl）, a low frequency tone （lft）, and atonal sound pulses （asp）, which were then followed by a frequency-modulated fall. Manipulating any of the rectrices （tail-feathers） of wild males impaired production of the lft and asp but not the hfl or fall, which are apparently vocal. I tested the sound production capabilities of the rectrices in a wind tunnel. Single rectrices could generate the lft but not the asp, whereas multiple rectrices tested together produced sounds sitlfilar to the asp when they fluttered and collided with their neighbors percussively, representing a previously unknown mechanism of sound production. During the shuttle display, a trill is generated by the wings during pulses in which the wingbeat frequency is elevated to 95 Hz, 40% higher than the typical hovering wingbeat frequency. The Calliope hummingbird courtship displays include sounds produced by three independent mechanisms, and thus include a minimum of three acoustic signal components. These acoustic mechanisms have different constraints and thus potentially contain different messages. Producing multiple acoustic signals via multiple mechanisms may be a way to escape the constraints present in any single mechanism .展开更多
基金The National Natural Science Foundation of China(No.10971025,10901035)
文摘For a graph G and two positive integers j and k an m-L j k -edge-labeling of G is an assignment from the set 0 1 … m-to the edges such that adjacent edges receive labels that differ by at least j and edges at distance two receive labels that differ by at least k.Theλ′j k-number of G denoted byλ′j k G is the minimum integer m overall m-L j k -edge-labeling of G.The necklace is a specific type of Halin graph.The L 1 2 -edge-labeling of necklaces is studied and the lower and upper bounds on λ′1 2-number for necklaces are given.Also both the lower and upper bounds are attainable.
基金The National Natural Science Foundation of China(No.10971025)
文摘Let j, k and m be three positive integers, a circular m-L（j, k）-labeling of a graph G is a mapping f： V（G）→{0, 1, …, m-1}such that f（u）-f（v）m≥j if u and v are adjacent, and f（u）-f（v）m≥k if u and v are at distance two,where a-bm=min{a-b,m-a-b}. The minimum m such that there exists a circular m-L（j, k）-labeling of G is called the circular L（j, k）-labeling number of G and is denoted by σj, k（G）. For any two positive integers j and k with j≤k,the circular L（j, k）-labeling numbers of trees, the Cartesian product and the direct product of two complete graphs are determined.
基金The National Natural Science Foundation of China(No10671033)Southeast University Science Foundation ( NoXJ0607230)
文摘L（ s, t）-labeling is a variation of graph coloring which is motivated by a special kind of the channel assignment problem. Let s and t be any two nonnegative integers. An L （s, t）-labeling of a graph G is an assignment of integers to the vertices of G such that adjacent vertices receive integers which differ by at least s, and vertices that are at distance of two receive integers which differ by at least t. Given an L（s, t） -labeling f of a graph G, the L（s, t） edge span of f, βst （ G, f） = max { ｜f（u） -f（v）｜： （ u, v） ∈ E（G） } is defined. The L（ s, t） edge span of G, βst（G）, is minβst（G,f）, where the minimum runs over all L（s, t）-labelings f of G. Let T be any tree with a maximum degree of △≥2. It is proved that if 2s≥t≥0, then βst（T） =（ [△/2 ] - 1）t ＋s; if 0≤2s 〈 t and △ is even, then βst（T） = [ （△ - 1） t/2 ] ; and if 0 ≤2s 〈 t and △ is odd, then βst（T） = （△ - 1） t/2 ＋ s. Thus, the L（s, t） edge spans of the Cartesian product of two paths and of the square lattice are completely determined.
基金Acknowledgments I thank S. Weinstein, A. Varma, and T. Feo for assistance in the field L. Benedict and T. Libby for use of equipment, J. Brown for accommodations, and G. Weston-Murphy for assistance with the wind tunnel. Walter Nussbaumer kindly allowed use of a photo. The manuscript was improved by comments from T. Feo and two anonymous reviewers. The research was supported by the MVZ and National Science Foundation IOS-090353 to R. Prum.
文摘Multi-component signals contain multiple signal parts expressed in the same physical modality. One way to identify individual components is if they are produced by different physical mechanisms. Here, I studied the mechanisms generating acoustic signals in the courtship displays of the Calliope hummingbird Stellula calliope. Display dives consisted of three synchronized sound elements, a high-frequency tone （hfl）, a low frequency tone （lft）, and atonal sound pulses （asp）, which were then followed by a frequency-modulated fall. Manipulating any of the rectrices （tail-feathers） of wild males impaired production of the lft and asp but not the hfl or fall, which are apparently vocal. I tested the sound production capabilities of the rectrices in a wind tunnel. Single rectrices could generate the lft but not the asp, whereas multiple rectrices tested together produced sounds sitlfilar to the asp when they fluttered and collided with their neighbors percussively, representing a previously unknown mechanism of sound production. During the shuttle display, a trill is generated by the wings during pulses in which the wingbeat frequency is elevated to 95 Hz, 40% higher than the typical hovering wingbeat frequency. The Calliope hummingbird courtship displays include sounds produced by three independent mechanisms, and thus include a minimum of three acoustic signal components. These acoustic mechanisms have different constraints and thus potentially contain different messages. Producing multiple acoustic signals via multiple mechanisms may be a way to escape the constraints present in any single mechanism .
