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The Number of Spanning Trees in Kn-Complements of Quasi-Threshold Graphs

Stavros D. Nikolopoulos, Charis Papadopoulos
2004 Graphs and Combinatorics  
In this paper we examine the classes of graphs whose K n -complements are trees or quasi-threshold graphs and derive formulas for their number of spanning trees; for a subgraph H of K n , the K n -complement  ...  Our results generalize previous results and extend the family of graphs of the form K n À H admitting formulas for the number of their spanning trees.  ...  Every node V i of the cent-tree T c ðQÞ contains exactly The Number of Spanning Trees in K n -Complements of Quasi-Threshold Graphs  ... 
doi:10.1007/s00373-004-0568-x fatcat:wtced4axqzatzhpm6okvhhi46e

The Number of Spanning Trees in Kn-complements of Quasi-threshold Graphs [article]

Stavros D. Nikolopoulos, Charis Papadopoulos
2005 arXiv   pre-print
In this paper we examine the classes of graphs whose K_n-complements are trees and quasi-threshold graphs and derive formulas for their number of spanning trees; for a subgraph H of K_n, the K_n-complement  ...  Our proofs are based on the complement spanning-tree matrix theorem, which expresses the number of spanning trees of a graph as a function of the determinant of a matrix that can be easily constructed  ...  Quasi-threshold Graphs In this section, we derive a formula for the number of the spanning trees of the graph K n − Q, where Q is a quasi-threshold graph.  ... 
arXiv:cs/0502038v1 fatcat:ynl2dp2jwrh7nduycl7kim2ghq

Laplacian Spectrum of Weakly Quasi-threshold Graphs

R. B. Bapat, A. K. Lal, Sukanta Pati
2008 Graphs and Combinatorics  
It turns out that weakly quasi-threshold graphs are Laplacian integral. As a corollary we obtain a closed formula for the number of spanning trees in such graphs.  ...  We show that weakly quasi-threshold graphs are precisely the comparability graphs of a forest consisting of rooted trees with each vertex of a tree being replaced by an independent set.  ...  We sincerely thank the referees for many helpful suggestions which has immensely improved the presentation of the article.  ... 
doi:10.1007/s00373-008-0785-9 fatcat:rlftjh33kvhlfdnjhzis33hbhy

Page 5170 of Mathematical Reviews Vol. , Issue 2004g [page]

2004 Mathematical Reviews  
Given a graph G, the spanning tree edge density, or hereafter simply density, of an edge e in G is the fraction of the spanning trees of G that contain e.  ...  Summary: “In the design of reliable and invulnerable networks, it is often a goal to maximize the number of spanning trees of a graph with a given number of vertices and edges.  ... 

On Characterizations for Subclasses of Directed Co-Graphs [article]

Frank Gurski, Dominique Komander, Carolin Rehs
2020 arXiv   pre-print
quasi threshold graphs.  ...  Among these are trivially perfect graphs, threshold graphs, weakly quasi threshold graphs, and simple co-graphs.  ...  n vertices, i.e. the complement graph of a complete directed graph on n vertices.  ... 
arXiv:1907.00801v2 fatcat:2d755okkgbfyhet6rki4adqw74

Maximizing the number of spanning trees in Kn-complements of asteroidal graphs

Stavros D. Nikolopoulos, Leonidas Palios, Charis Papadopoulos
2009 Discrete Mathematics  
In this paper we introduce the class of graphs whose complements are asteroidal (starlike) graphs and derive closed formulas for the number of spanning trees of its members.  ...  Additionally, we prove maximization theorems that enable us to characterize the graphs whose complements are asteroidal graphs and possess a maximum number of spanning trees.  ...  Acknowledgments The authors thank the anonymous referees whose suggestions helped improve the presentation of the paper.  ... 
doi:10.1016/j.disc.2008.08.008 fatcat:lhli4w2llzbsvil4lrxrqha3sq

Fast Diameter Computation within Split Graphs [article]

Guillaume Ducoffe
2021 arXiv   pre-print
graph in less than quadratic time – in the size n+m of the input.  ...  When can we compute the diameter of a graph in quasi linear time?  ...  Finally, the stabbing number of (X, R) is the minimum stabbing number over its spanning paths. Observation 2 For any split graph G = (K ∪ S, E), let S = {N G (u) | u ∈ S}.  ... 
arXiv:1910.03438v5 fatcat:kfmwqqhqe5a7blrimrt6qzgihq

Page 4720 of Mathematical Reviews Vol. , Issue 97H [page]

1997 Mathematical Reviews  
Let T,, be a tree of m vertices. The graph G studied in this paper is obtained by replacing each new edge of K, + T,, by a path of length n.  ...  97h:05158 quasi-threshold graph, and the class of quasi-threshold graphs is closed under disjoint union and under the operation of adding a new vertex adjacent to all vertices.  ... 

Minimal spanning forests

Russell Lyons,, Yuval Peres, Oded Schramm
2006 Annals of Probability  
In conjunction with a recent result of Gaboriau, this implies that in any Cayley graph, the expected degree of the FMSF is at least the expected degree of the FSF (the weak limit of uniform spanning trees  ...  Minimal spanning forests on infinite graphs are weak limits of minimal spanning trees from finite subgraphs.  ...  We thank Itai Benjamini for useful discussions at the early stages of this work, and Gábor Pete for comments on the manuscript.  ... 
doi:10.1214/009117906000000269 fatcat:fydt3y5qozcipaqdtwcmk5whbu

Quasi-threshold graphs

Yan Jing-Ho, Chen Jer-Jeong, Gerard J. Chang
1996 Discrete Applied Mathematics  
Quasi-threshold graphs are defined recursively by the following rules: (1) KI is a quasithreshold graph, (2) adding a new vertex adjacent to all vertices of a quasi-threshold graph results in a quasi-threshold  ...  graph, (3) the disjoint union of two quasi-threshold graphs is a quasithreshold graph.  ...  Acknowledgements The authors thank two anonymous referees for many constructive suggestions for a revision of this paper.  ... 
doi:10.1016/0166-218x(96)00094-7 fatcat:moznnihwvnf3vagtizx5kbxyum

A limit characterization for the number of spanning trees of graphs

S.D. Nikolopoulos, C. Nomikos, P. Rondogiannis
2004 Information Processing Letters  
In this paper we propose a limit characterization of the behaviour of classes of graphs with respect to their number of spanning trees.  ...  We study the spanning tree behaviour of the sequence {K n − G n } when n → ∞ and the number of edges of G n scales according to n.  ...  graph [1] , a multi-complete/star graph [3] , a quasi-threshold graph [9] , and so on (see Berge [1] for an exposition of the main results).  ... 
doi:10.1016/j.ipl.2004.03.001 fatcat:qqvn2gqabnczrkzhdajaemyiru

Enumerating threshold graphs and some related graph classes [article]

David Galvin, Greyson Wesley, Bailee Zacovic
2022 arXiv   pre-print
We also obtain an analog of the Frobenius formula (connecting Eulerian numbers and Stirling numbers of the second kind) in the context of labelled threshold graphs.  ...  We give combinatorial proofs of some enumeration formulas involving labelled threshold, quasi-threshold, loop-threshold and quasi-loop-threshold graphs.  ...  (k + 1) k−1 , ( 22 ) where qt n is the number of labelled quasi-threshold graphs on [n] .  ... 
arXiv:2110.08953v2 fatcat:hdzueugaanbbldt226prg37wvi

A survey of antiregular graphs

2020 Contributions to Mathematics  
The set of all different degrees of the vertices of a graph G is known as the degree set of G. A nontrivial graph of order n whose degree set consists of n−1 elements is called an antiregular graph.  ...  Antiregular graphs have been studied in literature also under other names, including "quasi-perfect graphs", "maximally nonregular graphs" and "degree antiregular graphs".  ...  The author would like to thank professors Gary Chartrand and Ping Zhang for their correspondence dealing with the topic of this paper.  ... 
doi:10.47443/cm.2020.0021 fatcat:hn4a4nwmjbf3bfitzdivqimmym

Matroid Intersections, Polymatroid Inequalities, and Related Problems [chapter]

Endre Boros, Khaled Elbassioni, Vladimir Gurvich, Leonid Khachiyan
2002 Lecture Notes in Computer Science  
Precisely, for a polymatroid function f and an integer threshold t ≥ 1, let α = α(f, t) denote the number of maximal sets X ⊆ V satisfying f (X) < t, let β = β(f, t) be the number of minimal sets X ⊆ V  ...  . , Mm on the common ground set V , it is shown that all maximal subsets of V , independent in the m matroids, can be generated in quasi-polynomial time.  ...  (If the n input graphs are just n disjoint edges, then B r−1 is the set of all spanning trees in the graph E 1 ∪ · · · ∪ E n , see [17] .)  ... 
doi:10.1007/3-540-45687-2_11 fatcat:ggtafactyngwvo5dusjol5ih34

Fast Diameter Computation within Split Graphs

Guillaume Ducoffe, Michel Habib, Laurent Viennot
2021 Discrete Mathematics & Theoretical Computer Science  
However, under SETH this cannot be done in truly subquadratic time for any $k = \omega(\log{n})$. - For the {\em complements} of $k$-clique-interval split graphs, we can compute their diameter in truly  ...  When can we compute the diameter of a graph in quasi linear time?  ...  Finally, the stabbing number of (X, R) is the minimum stabbing number over its spanning paths. Observation 2 For any split graph G = (K ∪ S, E), let S = {N G (u) | u ∈ S}.  ... 
doi:10.46298/dmtcs.6422 fatcat:qr7voo4ic5aaddhqnd7n3w5xl4
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