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Page 496 of Physical Review Vol. 75, Issue 3 [page]

1949 Physical Review  
There- fore the sum of the C(G) reduces to the single term C(G,), and again the disconnected graphs may be omitted from consideration.  ...  From G a “reduced graph” Go can be obtained by omit- ting F completely and joining the incoming line at x; with the outgoing line at x2 to form a single electron line in Go, the newly formed line being  ... 

Page 3038 of Mathematical Reviews Vol. , Issue 2002E [page]

2002 Mathematical Reviews  
Using this definition, the problem of determining the integral sum number and the sum number of the complete bipartite graphs K,,, which is an unsolved problem proposed by Harary, is investigated and solved  ...  It is clear that every sum graph has at least one isolated vertex. The sum number o(G) of the graph G is the least number of isolated vertices one must add to G to turn it into a sum graph.  ... 

The symmetric M-matrix and symmetric inverse M-matrix completion problems

Leslie Hogben
2002 Linear Algebra and its Applications  
diagonal block either complete or omitting all diagonal positions, or, in graph theoretic terms, if and only if every principal subpattern corresponding to a component of the graph of the pattern either  ...  omits all diagonal positions, or includes all positions. (3) A pattern has symmetric inverse M-completion if and only if its graph is block-clique and no diagonal position is omitted that corresponds  ...  A class of matrices is called a hereditary-sum-permutation-closed (HSP) class if: (1) every principal submatrix of a -matrix is a -matrix; (2) the direct sum of -matrices is a -matrix; (3) if A is a -matrix  ... 
doi:10.1016/s0024-3795(02)00301-4 fatcat:s6ahcsubyza6jgcyjkhwjcqhyy

Handicap Labelings of 4-Regular Graphs

Petr Kovar, Michal Kravcenko, Matej Krbecek, Adam Silber
2017 Advances in Electrical and Electronic Engineering  
Let G be a simple graph, let f  ...  SP2017/182 "Solving graph problems on spatiotemporal graphs with uncertainty using HPC", VSB-Technical University of Ostrava, Czech Republic.  ...  The work of the fourth author is partially supported by grant of SGS No.  ... 
doi:10.15598/aeee.v15i2.2263 fatcat:nu2ojsyeq5betmiyj5i7d5rllm

Sum graphs over all the integers

Frank Harary
1994 Discrete Mathematics  
Then a sum graph G is isomorphic to the sum graph of some ScN. This concept was discovered in [13], where some basic properties of the family 9 + of all sum graphs were presented.  ...  The sum number of a given graph G was defined as the smallest number of isolated nodes which when added to G result in a sum graph. The integral sum number of G is analogous.  ...  [S] , who noted that complete graphs are not 'mod sum graphs', i.e. cannot be realized as the sum graph of some ScZ,={O,1,2 )...) m-l} with addition modulo m.  ... 
doi:10.1016/0012-365x(92)00054-u fatcat:4tlxxfr2ubfobcs6sjodomutge

Counting complete matchings without using Pfaffians

N.G. de Bruijn
1980 Indagationes Mathematicae (Proceedings)  
It is known that the number of complete matchings of a planar graph can be expressed as a Pfaffian, and that the square of the Pfaffian of a matrix equals the determinant.  ...  In this paper it is shown without the use of Pfaffians that the square of the number of complete matchings equals that determinant.  ...  In any graph (V,E) with edge-weights bc (&E R) the sum of the weights of the complete matchings equals det B.  ... 
doi:10.1016/1385-7258(80)90037-2 fatcat:kkswzuv4e5eezkcvquugqikjuq

Linear Time Algorithms to the Minimum All-Ones Problem for Unicyclic and Bicyclic Graphs

William Y.C. Chen, Xueliang Li, Chao Wang, Xiaoyan Zhang
2004 Electronic Notes in Discrete Mathematics  
In this paper, we give graph-theoretic algorithms of linear time to the Minimum All-Ones Problem for unicyclic and bicyclic graphs.  ...  These algorithms are based on a graph-theoretic algorithm of linear time to the Minimum All-Ones Problem with Restrictions for trees.  ...  We omit the description and the proof of the algorithm for the Minimum Odd Sum Problem with Restrictions. The time complexity of the algorithm is linear.  ... 
doi:10.1016/j.endm.2004.03.018 fatcat:peu56s3tzbfexkrza4uh7fzjqm


2012 Journal of knot theory and its ramifications  
The determinant of this matrix after evaluation is shown to be the Jones polynomial of the pretzel knot by way of perfect matchings (or dimers) of this graph.  ...  The relationship is formalized between the familiar spanning tree setting for the Tait graph and the perfect matchings of the plane bipartite graph above.  ...  The terms in the permanent expansion of a bipartite adjacency submatrix associated with a balanced bipartite graph give the complete list of perfect matchings of the graph. Proof.  ... 
doi:10.1142/s0218216512500629 fatcat:my2pkcyjibejnkvlivoyltke2e

Page 6460 of Mathematical Reviews Vol. , Issue 96k [page]

1996 Mathematical Reviews  
If there is a countable universal graph for the class of graphs omitting C, then C = {C3,Cs,---, Ca 41}, ie. C contains only cycles of odd lengths > 3. Walter M.  ...  It is also known that for every positive integer k there exists a universal countable graph for the class of graphs omitting odd cycles of lengths at most 2k + 1.  ... 

Entangled Graphs [article]

Hadi Rahiminia, Massoud Amini
2006 arXiv   pre-print
We also show that the density matrix of a graph with only one entangled edge is entangled.  ...  The authors would like to thank Professor Braunstein and his graduate students for their kind help during the preparation of the original version of this paper.  ...  We omit all the entangled edges of graph G except P [ 1 √ 2 (| ij − | st )] and call the resulting graph H. We consider the vertex X as in Theorem 1.  ... 
arXiv:cs/0609156v1 fatcat:wwktu35445d5bbyovfrdejwg6i

Page 53 of Mathematical Reviews Vol. , Issue 99a [page]

1991 Mathematical Reviews  
These authors found that finite quasi-median graphs are precisely the retracts of finite Hamming graphs (i.e., products of complete graphs).  ...  Graphs omitting a bushy tree. (English summary) J. Graph Theory 26 (1997), no. 4, 203-210. Summary: “A tree is called bushy if it has no vertex of degree 2.  ... 

Magic powers of graphs

Marián Trenkler, Vladimír Vetchý
1997 Mathematica Bohemica  
Since, except of the complete graph K-2 of order 2, no graph with less than 5 vertices is magic we confine ourselves to graphs of order n ^ 5.  ...  By a magic valuation of a graph G we mean such an assignment of the edges of G by pairwise different positive numbers that the sum of assignments of edges meeting the same vertex is constant.  ... 
doi:10.21136/mb.1997.125922 fatcat:icbt37o3obbkhpyfnsznhxu3qu

On the mean chromatic number

Martin Anthony
1994 Discrete Mathematics  
The mean chromatic number of a graph is a measure of the expected performance of the greedy vertex-colouring algorithm when each ordering of the vertices is equally likely.  ...  Some results on the value of the mean chromatic number and its asymptotic behaviour are presented.  ...  We remark that in any greedy colouring of the complete bipartite graph K,,, (and indeed, of any complete bipartite graph), exactly two colours are used. Hence, for any n, ii(K,,,)=2.  ... 
doi:10.1016/0012-365x(94)90139-2 fatcat:ck577fuz7zgddnanui7bnxam6m

Page 4536 of Mathematical Reviews Vol. , Issue 96h [page]

1996 Mathematical Reviews  
The four exceptions are the classes of (1) one-vertex graphs, (2) discrete graphs, (3) disjoint sums of one-edge graphs and one- vertex graphs, and (4) disjoint sums of complete bi-partite graphs and one-vertex  ...  In the second, he gives an interesting necessary and sufficient condition for omitting an arbitrary set of complete types.  ... 

Sum Multi-coloring of Graphs [chapter]

Amotz Bar-Noy, Magnús M. Halldórsson, Guy Kortsarz, Ravit Salman, Hadas Shanhnai
1999 Lecture Notes in Computer Science  
Scheduling dependent jobs on multiple machines is modeled by the graph multi-coloring problem. In this paper, we consider the problem of minimizing the average completion time of all jobs.  ...  This paper reports a comprehensive study of the sum multi-coloring problem. We extend most of the known results for the sum coloring problem into the three variants of the sum multicoloring problem.  ...  Then minimizing the average completion time is equivalent to minimizing the sum of all these numbers.  ... 
doi:10.1007/3-540-48481-7_34 fatcat:grw6hvyiujaz7c3vjxa4bqym74
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