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New types of integral sum graphs

Jianqiang Wu, Jingzhong Mao, Deying Li
2003 Discrete Mathematics  
In this paper we enlarge the 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.  ... 
doi:10.1016/s0012-365x(02)00456-9 fatcat:kyge6z3szjb7lkxtjc2y3o6cue

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].  ... 

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.  ... 

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  ... 

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

J. Gillis, J. Jedwab, D. Zeilberger
1988 SIAM Journal on Mathematical Analysis  
In addition, a number 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).  ... 
doi:10.1137/0519109 fatcat:adackwudanf63p6vka34vh2gta

Macdonald Polynomials and Chromatic Quasisymmetric Functions

James Haglund, Andrew Timothy Wilson
2020 Electronic Journal of Combinatorics  
We express the 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).  ... 
doi:10.37236/9011 fatcat:i227miog6bgsppbut24y62k57e

Integral sum graphs from identification

Zhibo Chen
1998 Discrete Mathematics  
In this paper, we first introduce some methods 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.  ... 
doi:10.1016/s0012-365x(97)00046-0 fatcat:rsliaguf75fcxiyxqsxlz4vcuq

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]

R. Y. Kamath, W. R. Cyre
1995 Lecture Notes in Computer Science  
This paper describes a rule-based algorithm for the 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.  ... 
doi:10.1007/3-540-60161-9_28 fatcat:eikwnhxgj5atthhbksgveo32ke

The ne plus ultra of tree graph inequalities

Paul Federbush
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.  ... 
doi:10.1007/bf00429953 fatcat:w3kvljp475fy7kruhk74wlnxqu

Running Nonlocal Lagrangians [article]

V. Bhansali, H. Georgi
1992 arXiv   pre-print
In a scalar field theory, we write an integro-differential renormalization group equation for every possible class 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  ... 
arXiv:hep-ph/9205242v1 fatcat:6rek54zprnbyldzk4nt4uf4g2y

Macdonald polynomials and chromatic quasisymmetric functions [article]

James Haglund, Andrew Timothy Wilson
2018 arXiv   pre-print
We express the 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.  ... 
arXiv:1701.05622v2 fatcat:cdexduugmvfjtp6zqnirlrt36a

RenormalizedS-Matrix for Scalar Electrodynamics

Abdus Salam
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-  ... 
doi:10.1103/physrev.86.731 fatcat:usl4zpaybverhg6vfztu4kea64

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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