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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 ...##
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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

2002
*
Linear Algebra and its Applications
*

diagonal block either

doi:10.1016/s0024-3795(02)00301-4
fatcat:s6ahcsubyza6jgcyjkhwjcqhyy
*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 ...##
###
Handicap Labelings of 4-Regular Graphs

2017
*
Advances in Electrical and Electronic Engineering
*

Let G be a simple

doi:10.15598/aeee.v15i2.2263
fatcat:nu2ojsyeq5betmiyj5i7d5rllm
*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. ...##
###
Sum graphs over all the integers

1994
*
Discrete Mathematics
*

Then a

doi:10.1016/0012-365x(92)00054-u
fatcat:4tlxxfr2ubfobcs6sjodomutge
*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. ...##
###
Counting complete matchings without using Pfaffians

1980
*
Indagationes Mathematicae (Proceedings)
*

It is known that the number

doi:10.1016/1385-7258(80)90037-2
fatcat:kkswzuv4e5eezkcvquugqikjuq
*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. ...##
###
Linear Time Algorithms to the Minimum All-Ones Problem for Unicyclic and Bicyclic Graphs

2004
*
Electronic Notes in Discrete Mathematics
*

In this paper, we give

doi:10.1016/j.endm.2004.03.018
fatcat:peu56s3tzbfexkrza4uh7fzjqm
*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. ...##
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A DETERMINANT FORMULA FOR THE JONES POLYNOMIAL OF PRETZEL KNOTS

2012
*
Journal of knot theory and its ramifications
*

The determinant

doi:10.1142/s0218216512500629
fatcat:my2pkcyjibejnkvlivoyltke2e
*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. ...##
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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]

2006
*
arXiv
*
pre-print

We also show that the density matrix

arXiv:cs/0609156v1
fatcat:wwktu35445d5bbyovfrdejwg6i
*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. ...##
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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

1997
*
Mathematica Bohemica
*

Since, except

doi:10.21136/mb.1997.125922
fatcat:icbt37o3obbkhpyfnsznhxu3qu
*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. ...##
###
On the mean chromatic number

1994
*
Discrete Mathematics
*

The mean chromatic number

doi:10.1016/0012-365x(94)90139-2
fatcat:ck577fuz7zgddnanui7bnxam6m
*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. ...##
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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. ...##
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Sum Multi-coloring of Graphs
[chapter]

1999
*
Lecture Notes in Computer Science
*

Scheduling dependent jobs on multiple machines is modeled by the

doi:10.1007/3-540-48481-7_34
fatcat:grw6hvyiujaz7c3vjxa4bqym74
*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. ...
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