scholarly journals Spanning Trees and Function Classes

10.37236/1650 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Jeffery B. Remmel ◽  
S. Gill Williamson
Keyword(s):  
Spanning Trees ◽  
Directed Graphs ◽  
Complete Multipartite Graph ◽  
Natural Class ◽  
Function Classes ◽  
Cayley Trees ◽  
Spanning Forests ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
And Function

If $G=K_n$ is the complete graph, the classical Prüffer correspondence gives a natural bijection between all spanning trees of $G$ (i.e., all Cayley trees) and all functions from a set of $n-2$ elements to a set of $n$ elements. If $G$ is a complete multipartite graph, then such bijections have been studied by Eğecioğlu and Remmel. In this paper, we define a class of directed graphs, called filtered digraphs, and describe a natural class of bijections between oriented spanning forests of these digraphs and associated classes of functions. We derive multivariate generating functions for the oriented spanning forests which arise in this context, and we link basic properties of these spanning forests to properties of the functions to which they correspond. This approach yields a number of new results for directed graphs. Moreover, in the undirected case, various specializations of our multivariate generating function not only include various known results but also give a number of new results.

A CLASS OF GRAPHS WHICH HAS EFFICIENT RANKING AND UNRANKING ALGORITHMS FOR SPANNING TREES AND FORESTS

2004 ◽  
Vol 15 (04) ◽  
pp. 619-648
Author(s):  
ÖMER EĞECIOĞLU ◽  
JEFFREY B. REMMEL ◽  
S. GILL WILLIAMSON
Keyword(s):  
Generating Functions ◽  
Spanning Trees ◽  
Directed Graphs ◽  
Efficient Algorithms ◽  
Explicit Formulas ◽  
Natural Class ◽  
Spanning Forests

Remmel and Williamson recently defined a class of directed graphs, called filtered digraphs, and described a natural class of bijections between oriented spanning forests of these digraphs and associated classes of functions [12]. Filtered digraphs include many specialized graphs such as complete k-partite graphs. The Remmel-Williamson bijections provide explicit formulas for various multivariate generating functions for the oriented spanning forests which arise in this context. In this paper, we prove another important property of these bijections, namely, that it allows one to construct efficient algorithms for ranking and unranking spanning trees or spanning forests of filtered digraphs G. For example, we show that if G=(V,E) is a filtered digraph and SP(G) is the collection of spanning trees of G, then our algorithm requires O(|V|) operations of sum, difference, product, quotient, and comparison of numbers less than or equal |SP(G)| to rank or unrank spanning trees of G.

scholarly journals The number of spanning trees of a complete multipartite graph

1999 ◽  
Vol 197-198 ◽  
pp. 537-541 ◽  
Author(s):  
Richard P. Lewis
Keyword(s):  
Spanning Trees ◽  
Complete Multipartite Graph ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Number Of Spanning Trees

scholarly journals On the number of spanning trees of K_n^m #x00B1 G graphs

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Stavros D. Nikolopoulos ◽  
Charis Papadopoulos
Keyword(s):  
Complete Graph ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Star Graph ◽  
Complete Multipartite Graph ◽  
The Family ◽  
International Audience ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Number Of Spanning Trees

International audience The K_n-complement of a graph G, denoted by K_n-G, is defined as the graph obtained from the complete graph K_n by removing a set of edges that span G; if G has n vertices, then K_n-G coincides with the complement øverlineG of the graph G. In this paper we extend the previous notion and derive determinant based formulas for the number of spanning trees of graphs of the form K_n^m #x00b1 G, where K_n^m is the complete multigraph on n vertices with exactly m edges joining every pair of vertices and G is a multigraph spanned by a set of edges of K_n^m; the graph K_n^m + G (resp. K_n^m - G) is obtained from K_n^m by adding (resp. removing) the edges of G. Moreover, we derive determinant based formulas for graphs that result from K_n^m by adding and removing edges of multigraphs spanned by sets of edges of the graph K_n^m. We also prove closed formulas for the number of spanning tree of graphs of the form K_n^m #x00b1 G, where G is (i) a complete multipartite graph, and (ii) a multi-star graph. Our results generalize previous results and extend the family of graphs admitting formulas for the number of their spanning trees.

Complete Multipartite Graph Codes

2018 ◽  
Author(s):  
Yuta Kumano ◽  
Yoshiyuki Sakamaki ◽  
Hironori Uchikawa
Keyword(s):  
Complete Multipartite Graph ◽  
Graph Codes ◽  
Multipartite Graph ◽  
Complete Multipartite

scholarly journals The Size, Multipartite Ramsey Numbers for nK2 Versus Path–Path and Cycle

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 764
Author(s):  
Yaser Rowshan ◽  
Mostafa Gholami ◽  
Stanford Shateyi
Keyword(s):  
Positive Integer ◽  
Ramsey Number ◽  
Complete Multipartite Graph ◽  
Ramsey Numbers ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Monochromatic Copy

For given graphs G1,G2,…,Gn and any integer j, the size of the multipartite Ramsey number mj(G1,G2,…,Gn) is the smallest positive integer t such that any n-coloring of the edges of Kj×t contains a monochromatic copy of Gi in color i for some i, 1≤i≤n, where Kj×t denotes the complete multipartite graph having j classes with t vertices per each class. In this paper, we computed the size of the multipartite Ramsey numbers mj(K1,2,P4,nK2) for any j,n≥2 and mj(nK2,C7), for any j≤4 and n≥2.

Codes from multipartite graphs and minimal permutation decoding sets

2015 ◽  
Vol 07 (04) ◽  
pp. 1550060
Author(s):  
P. Seneviratne
Keyword(s):  
Automorphism Group ◽  
Error Correcting Code ◽  
Binary Codes ◽  
Hard Problem ◽  
Complete Multipartite Graph ◽  
Decoding Algorithm ◽  
Permutation Decoding ◽  
Multipartite Graphs ◽  
Multipartite Graph ◽  
Complete Multipartite

Permutation decoding method developed by MacWilliams and described in [Permutation decoding of systematic codes, Bell Syst. Tech. J. 43 (1964) 485–505] is a decoding technique that uses a subset of the automorphism group of the code called a PD-set. The complexity of the permutation decoding algorithm depends on the size of the PD-set and finding a minimal PD-set for an error correcting code is a hard problem. In this paper we examine binary codes from the complete-multipartite graph [Formula: see text] and find PD-sets for all values of [Formula: see text] and [Formula: see text]. Further we show that these PD-sets are minimal when [Formula: see text] is odd and [Formula: see text].

scholarly journals The Energy Change of the Complete Multipartite Graph

2020 ◽  
Vol 36 (36) ◽  
pp. 309-317
Author(s):  
Haiying Shan ◽  
Changxiang He ◽  
Zhensheng Yu
Keyword(s):  
Linear Algebra ◽  
Singular Values ◽  
New Method ◽  
Energy Change ◽  
Natural Question ◽  
Complete Multipartite Graph ◽  
Multipartite Graph ◽  
Energy Of Graph ◽  
Complete Multipartite ◽  
Edge Addition

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. Akbari et al. [S. Akbari, E. Ghorbani, and M. Oboudi. Edge addition, singular values, and energy of graphs and matrices. {\em Linear Algebra Appl.}, 430:2192--2199, 2009.] proved that for a complete multipartite graph $K_{t_1 ,\ldots,t_k}$, if $t_i\geq 2 \ (i=1,\ldots,k)$, then deleting any edge will increase the energy. A natural question is how the energy changes when $\min\{t_1 ,\ldots,t_k\}=1$. In this paper, a new method to study the energy of graph is explored. As an application of this new method, the above natural question is answered and it is completely determined how the energy of a complete multipartite graph changes when one edge is removed.

scholarly journals On the euclidean dimension of a complete multipartite graph

1988 ◽  
Vol 72 (1-3) ◽  
pp. 285-289 ◽  
Author(s):  
Hiroshi Maehara
Keyword(s):  
Complete Multipartite Graph ◽  
Multipartite Graph ◽  
Complete Multipartite
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