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New types of integral sum graphs

2003
*
Discrete Mathematics
*

In this paper we enlarge the

doi:10.1016/s0012-365x(02)00456-9
fatcat:kyge6z3szjb7lkxtjc2y3o6cue
*types**of**integral**sum**graphs*. We prove that some caterpillars and mPn are*integral**sum**graphs*and some conjectures*of*Harary are wrong. ... Harary (Discrete Math. 124(1-3) (1994) 99) introduced the intergral*sum**graphs*, and proved that the path Pn and matching mK2 are integer*sum**graphs*, and o ered some conjectures. ... Acknowledgements The authors wish to thank the referee for useful comments and suggestions that have allowed them to improve the presentation*of*this paper. ...##
###
Page 7570 of Mathematical Reviews Vol. , Issue 98M
[page]

1998
*
Mathematical Reviews
*

In the present paper the author first introduces some methods

*of*identification for constructing*new*connected*integral**sum**graphs*from given*integral**sum**graphs*. ... The idea*of**integral**sum**graphs*was introduced by F. Harary [Discrete Math. 124 (1994), no. 1-3, 99-105; MR 95b:05202]. ...##
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Page 3605 of Mathematical Reviews Vol. , Issue 94f
[page]

1994
*
Mathematical Reviews
*

Metric invariants, named topological indices in chemical literature, such as the

*graph*- distance index (the total*sum**of*all distances in the*graph*G), the eccentricity index (the*sum**of*eccentricies*of*... vertices in G), the*graph*chain-distance index (the*sum**of*chain-distances in G) and the chain-eccentricity index (the*sum**of*chain-eccentricities*of*vertices in G) are defined. ...##
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Page 187 of Physical Review Vol. 94, Issue 1
[page]

1954
*
Physical Review
*

When the summation over all

*graphs*is made, there occur a set*of**graphs*all*of*which have the same reduced*integral*and the same*type**of*subpart (vertex or self- energy), but in the set this part appears ... they cancel exactly with similar terms from the P brackets in the S matrix.® The*new*field variables ¥,*, ¥:, and A; are operators which annihilate and create a*new**type**of*bare particles which have not ...##
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Page 8955 of Mathematical Reviews Vol. , Issue 2003m
[page]

2003
*
Mathematical Reviews
*

8955
The

*integral**sum**graph*G*(S) is defined just as the*sum**graph*, the difference being that S Cc Z instead*of*SCN, Z= {...,—-3, —2,—1,0,1,2,3,...}. ... Harary [Discrete Math. 124 (1994), no. 1-3, 99-105; MR 95b:05202] has pointed out that, for each positive integer n, the path P,, and matching nK> are*integral**sum**graphs*. ...##
###
A Combinatorial Interpretation of the Integral of the Product of Legendre Polynomials

1988
*
SIAM Journal on Mathematical Analysis
*

In addition, a number

doi:10.1137/0519109
fatcat:adackwudanf63p6vka34vh2gta
*of*apparently*new*combinatorial identities are obtained. ... k, not all*of*which seem analytically straightforward, including a direct combinatorial derivation*of*the known formula for Ia,b, and the expression*of*Ia,b,c,d as a simple finite*sum*. ... It remains to determine how many such*graphs*there are. Consider (12). Since there are B edges*of**type*1 3 in the*graph*, this will also be the number*of*segments*of**type*(ii). ...##
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Macdonald Polynomials and Chromatic Quasisymmetric Functions

2020
*
Electronic Journal of Combinatorics
*

We express the

doi:10.37236/9011
fatcat:i227miog6bgsppbut24y62k57e
*integral*form Macdonald polynomials as a weighted*sum**of*Shareshian and Wachs' chromatic quasisymmetric functions*of*certain*graphs*. ... Then we use known expansions*of*these chromatic quasisymmetric functions into Schur and power*sum*symmetric functions to provide Schur and power*sum*formulas for the*integral*form Macdonald polynomials ... We let IFT µ denote the collection*of*all*integral*form tableaux*of**type*µ (*of*varying shapes λ). is an*integral*form tableau*of**type*(2, 2, 2) and shape (3, 2, 1). ...##
###
Integral sum graphs from identification

1998
*
Discrete Mathematics
*

In this paper, we first introduce some methods

doi:10.1016/s0012-365x(97)00046-0
fatcat:rsliaguf75fcxiyxqsxlz4vcuq
*of*identification on constructing*new*connected*integral**sum**graphs*from given*integral**sum**graphs*. ... The idea*of**integral**sum**graphs*was introduced by Harary (1994) . ... In the present paper, we shall present some methods on constructing*new*connected*integral**sum**graphs*from given*integral**sum**graphs*by identification. ...##
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Page 531 of Physical Review Vol. 88, Issue 3
[page]

1952
*
Physical Review
*

Any ¢ may be replaced by a contribution in- volving a vertex

*of**type*(0’), yielding a*new**graph*which must be considered in the enumeration. ... Letting A[@o ] be the contribution to the energy arising from*graphs*in 2’, the condition that the*sum**of*all contributions from subgraphs*of**type*(8) vanishes is then expressed by the equation 5A[ Go ...##
###
Automatic integration of digital system requirements using schemata
[chapter]

1995
*
Lecture Notes in Computer Science
*

This paper describes a rule-based algorithm for the

doi:10.1007/3-540-60161-9_28
fatcat:eikwnhxgj5atthhbksgveo32ke
*integration**of*requirements which are expressed as conceptual*graphs*. ... Natural language and diagrams*of*various*types*are commonly used for specifications on digital systems containing descriptions and requirements. ... Whenever the*integrator*comes across a*new*concept*type*in the requirement*graph*it searches the database for a schema for the concept*type*. ...##
###
The ne plus ultra of tree graph inequalities

1990
*
Letters in Mathematical Physics
*

*of*the interaction, and

*integration*by parts. ... This includes applications to both two-body and many-body potential situations

*of*the recently obtained '

*new*improved tree

*graph*inequalities" that have led to the 'extra I/N! factors'. ... by a

*sum*

*of*terms

*of*similar

*type*. ...

##
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Running Nonlocal Lagrangians
[article]

1992
*
arXiv
*
pre-print

In a scalar field theory, we write an integro-differential renormalization group equation for every possible class

arXiv:hep-ph/9205242v1
fatcat:6rek54zprnbyldzk4nt4uf4g2y
*of**graph*at one loop order. ... We investigate the renormalization*of*"nonlocal" interactions in an effective field theory using dimensional regularization with minimal subtraction. ... Hence the resummation*of*the pole terms for a*sum*starting at zero gives a higher dimensional angular*integral*. 2m-point function*of**type*n at one loop For the non-trivial general*graph*with n ≥ 2, after ...##
###
Macdonald polynomials and chromatic quasisymmetric functions
[article]

2018
*
arXiv
*
pre-print

We express the

arXiv:1701.05622v2
fatcat:cdexduugmvfjtp6zqnirlrt36a
*integral*form Macdonald polynomials as a weighted*sum**of*Shareshian and Wachs' chromatic quasisymmetric functions*of*certain*graphs*. ... Then we use known expansions*of*these chromatic quasisymmetric functions into Schur and power*sum*symmetric functions to provide Schur and power*sum*formulas for the*integral*form Macdonald polynomials ... First, we define a*new*weight function on*integral*form tableaux. To compute wt J (α) µ ′ (x) = T ∈IFTµ wt (α) µ (T )s shape(T ) (54) where the*sum*is over all*integral*form tableaux*of**type*µ. ...##
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RenormalizedS-Matrix for Scalar Electrodynamics

1952
*
Physical Review
*

*new*

*type*

*of*“vertex,” illustrated in Fig. 5 with the vertex factor 2ip, since [—1/2mi(0/0p,)Dr=Dr-2ip,: Dy» |. ... Yy»* similarly denotes the

*sum*

*of*

*integrals*for all

*graphs*with only the first and the last end-vertices unchanged 4-vertices, and Z,»* the

*sum*for

*graphs*with only the first end-vertex an unchanged 4- ...

##
###
Page 612 of Physical Review Vol. 83, Issue 3
[page]

1951
*
Physical Review
*

Now, whenever Y; has one

*of*the last four forms (58), the corresponding vertex x; is*of*a*new**type*, having two photon lines and no electron line incident. ... The coefficient M is a*sum**of*contributions Mz from the various terms in the*sum*(55). ...
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