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

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*...

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

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*. ...

##
###
Laplacian Spectrum of Weakly Quasi-threshold Graphs

2008
*
Graphs and Combinatorics
*

It turns out that weakly

doi:10.1007/s00373-008-0785-9
fatcat:rlftjh33kvhlfdnjhzis33hbhy
*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. ...##
###
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]

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. ...

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

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. ...

##
###
Fast Diameter Computation within Split Graphs
[article]

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}. ...

##
###
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

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. ...

##
###
Quasi-threshold graphs

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. ...

##
###
A limit characterization for the number of spanning trees of graphs

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). ...

##
###
Enumerating threshold graphs and some related graph classes
[article]

2022
*
arXiv
*
pre-print

We also obtain an analog

arXiv:2110.08953v2
fatcat:hdzueugaanbbldt226prg37wvi
*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*] . ...##
###
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. ...

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

2002
*
Lecture Notes in Computer Science
*

Precisely, for a polymatroid function f and an integer

doi:10.1007/3-540-45687-2_11
fatcat:ggtafactyngwvo5dusjol5ih34
*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] .) ...##
###
Fast Diameter Computation within Split Graphs

2021
*
Discrete Mathematics & Theoretical Computer Science
*

However, under SETH this cannot be done

doi:10.46298/dmtcs.6422
fatcat:qr7voo4ic5aaddhqnd7n3w5xl4
*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}. ...
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