scholarly journals On the Number of Spanning Trees in Random Regular Graphs

10.37236/3752 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Catherine Greenhill ◽  
Matthew Kwan ◽  
David Wind
Keyword(s):  
Asymptotic Distribution ◽  
Regular Graph ◽  
Spanning Trees ◽  
Regular Graphs ◽  
Cubic Graph ◽  
Expected Number ◽  
Fixed Integer ◽  
Numerical Evidence ◽  
Number Of Spanning Trees

Let $d\geq 3$ be a fixed integer.   We give an asympotic formula for the expected number of spanning trees in a uniformly random $d$-regular graph with $n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if $d$ is odd.) We also obtain the asymptotic distribution of the number of spanning trees in a uniformly random cubic graph, and conjecture that the corresponding result holds for arbitrary (fixed) $d$. Numerical evidence is presented which supports our conjecture.

scholarly journals On the Resistance Matrix of a Graph

10.37236/5295 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Jiang Zhou ◽  
Zhongyu Wang ◽  
Changjiang Bu
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Spanning Trees ◽  
Regular Graphs ◽  
Resistance Distance ◽  
Strongly Regular ◽  
Resistance Matrix ◽  
Number Of Spanning Trees

Let $G$ be a connected graph of order $n$. The resistance matrix of $G$ is defined as $R_G=(r_{ij}(G))_{n\times n}$, where $r_{ij}(G)$ is the resistance distance between two vertices $i$ and $j$ in $G$. Eigenvalues of $R_G$ are called R-eigenvalues of $G$. If all row sums of $R_G$ are equal, then $G$ is called resistance-regular. For any connected graph $G$, we show that $R_G$ determines the structure of $G$ up to isomorphism. Moreover, the structure of $G$ or the number of spanning trees of $G$ is determined by partial entries of $R_G$ under certain conditions. We give some characterizations of resistance-regular graphs and graphs with few distinct R-eigenvalues. For a connected regular graph $G$ with diameter at least $2$, we show that $G$ is strongly regular if and only if there exist $c_1,c_2$ such that $r_{ij}(G)=c_1$ for any adjacent vertices $i,j\in V(G)$, and $r_{ij}(G)=c_2$ for any non-adjacent vertices $i,j\in V(G)$.

scholarly journals Spanning trees in random regular uniform hypergraphs

2021 ◽  
pp. 1-25
Author(s):  
Catherine Greenhill ◽  
Mikhail Isaev ◽  
Gary Liang
Keyword(s):  
Asymptotic Distribution ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Threshold Value ◽  
Regular Graphs ◽  
Cubic Graphs ◽  
Trivial Case ◽  
Uniform Hypergraphs ◽  
Vertex Set ◽  
Number Of Spanning Trees

Abstract Let $${{\mathcal G}_{n,r,s}}$$ denote a uniformly random r-regular s-uniform hypergraph on the vertex set {1, 2, … , n}. We establish a threshold result for the existence of a spanning tree in $${{\mathcal G}_{n,r,s}}$$ , restricting to n satisfying the necessary divisibility conditions. Specifically, we show that when s ≥ 5, there is a positive constant ρ(s) such that for any r ≥ 2, the probability that $${{\mathcal G}_{n,r,s}}$$ contains a spanning tree tends to 1 if r > ρ(s), and otherwise this probability tends to zero. The threshold value ρ(s) grows exponentially with s. As $${{\mathcal G}_{n,r,s}}$$ is connected with probability that tends to 1, this implies that when r ≤ ρ(s), most r-regular s-uniform hypergraphs are connected but have no spanning tree. When s = 3, 4 we prove that $${{\mathcal G}_{n,r,s}}$$ contains a spanning tree with probability that tends to 1, for any r ≥ 2. Our proof also provides the asymptotic distribution of the number of spanning trees in $${{\mathcal G}_{n,r,s}}$$ for all fixed integers r, s ≥ 2. Previously, this asymptotic distribution was only known in the trivial case of 2-regular graphs, or for cubic graphs.

ON THE NORMALISED LAPLACIAN SPECTRUM, DEGREE-KIRCHHOFF INDEX AND SPANNING TREES OF GRAPHS

2015 ◽  
Vol 91 (3) ◽  
pp. 353-367 ◽  
Author(s):  
JING HUANG ◽  
SHUCHAO LI
Keyword(s):  
Lower Bound ◽  
Characteristic Polynomial ◽  
Regular Graph ◽  
Spanning Trees ◽  
Line Graph ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Subdivision Graph ◽  
Number Of Spanning Trees ◽  
Sharp Lower Bound

Given a connected regular graph $G$, let $l(G)$ be its line graph, $s(G)$ its subdivision graph, $r(G)$ the graph obtained from $G$ by adding a new vertex corresponding to each edge of $G$ and joining each new vertex to the end vertices of the corresponding edge and $q(G)$ the graph obtained from $G$ by inserting a new vertex into every edge of $G$ and new edges joining the pairs of new vertices which lie on adjacent edges of $G$. A formula for the normalised Laplacian characteristic polynomial of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$) in terms of the normalised Laplacian characteristic polynomial of $G$ and the number of vertices and edges of $G$ is developed and used to give a sharp lower bound for the degree-Kirchhoff index and a formula for the number of spanning trees of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$).

scholarly journals Smallest regular graphs without near 1-factors

1986 ◽  
Vol 41 (2) ◽  
pp. 193-210 ◽  
Author(s):  
Gary Chartrand ◽  
Sergio Ruiz ◽  
Curtiss E. Wall
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Regular Graphs ◽  
Cubic Graph ◽  
Subgraph Isomorphic ◽  
Even Order

AbstractA near 1-factor of a graph of order 2n ≧ 4 is a subgraph isomorphic to (n − 2) K2 ∪ P3 ∪ K1. Wallis determined, for each r ≥ 3, the order of a smallest r-regular graph of even order without a 1-factor; while for each r ≧ 3, Chartrand, Goldsmith and Schuster determined the order of a smallest r-regular, (r − 2)-edge-connected graph of even order without a 1-factor. These results are extended to graphs without near 1-factors. It is known that every connected, cubic graph with less than six bridges has a near 1-factor. The order of a smallest connected, cubic graph with exactly six bridges and no near 1-factor is determined.

scholarly journals Perfect roman domination in regular graphs

2018 ◽  
Vol 12 (1) ◽  
pp. 143-152 ◽  
Author(s):  
Michael Henning ◽  
William Klostermeyer
Keyword(s):  
Regular Graph ◽  
Minimum Weight ◽  
Domination Number ◽  
Regular Graphs ◽  
Cubic Graph ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Dominating Function

A perfect Roman dominating function on a graph G is a function f : V (G) ? {0,1,2} satisfying the condition that every vertex u with f(u) = 0 is adjacent to exactly one vertex v for which f(v) = 2. The weight of a perfect Roman dominating function f is the sum of the weights of the vertices. The perfect Roman domination number of G, denoted ?pR(G), is the minimum weight of a perfect Roman dominating function in G. We show that if G is a cubic graph on n vertices, then ?pR(G) ? 3/4n, and this bound is best possible. Further, we show that if G is a k-regular graph on n vertices with k at least 4, then ?pR(G) ? (k2+k+3/k2+3k+1)n.

Heuristic maximization of the number of spanning trees in regular graphs

2006 ◽  
Vol 343 (3) ◽  
pp. 309-325
Author(s):  
Jacek Wojciechowski ◽  
Jarosław Arabas ◽  
Błażej Sawionek
Keyword(s):  
Spanning Trees ◽  
Regular Graphs ◽  
Number Of Spanning Trees

scholarly journals Subgraph counts for dense random graphs with specified degrees

2020 ◽  
pp. 1-38
Author(s):  
Catherine Greenhill ◽  
Mikhail Isaev ◽  
Brendan D. McKay
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Spanning Trees ◽  
Degree Sequence ◽  
Complex Function ◽  
Random Permutation ◽  
Expected Number ◽  
Asymptotic Expressions ◽  
Number Of Spanning Trees

Abstract We prove two estimates for the expectation of the exponential of a complex function of a random permutation or subset. Using this theory, we find asymptotic expressions for the expected number of copies and induced copies of a given graph in a uniformly random graph with degree sequence(d 1 , …, d n ) as n→ ∞. We also determine the expected number of spanning trees in this model. The range of degrees covered includes d j = λn + O(n1/2+ε) for some λ bounded away from 0 and 1.

scholarly journals Rainbow Connection of Sparse Random Graphs

10.37236/2784 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Alan Frieze ◽  
Charalampos E. Tsourakakis
Keyword(s):  
Random Graphs ◽  
Regular Graph ◽  
Connected Graph ◽  
High Probability ◽  
Regular Graphs ◽  
Colored Graph ◽  
Fixed Integer ◽  
Rainbow Connection ◽  
Connectivity Threshold

An edge colored graph $G$ is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this work we study the rainbow connectivity of binomial random graphs at the connectivity threshold $p=\frac{\log n+\omega}{n}$ where $\omega=\omega(n)\to\infty$ and ${\omega}=o(\log{n})$ and of random $r$-regular graphs where $r \geq 3$ is a fixed integer. Specifically, we prove that the rainbow connectivity $rc(G)$ of $G=G(n,p)$ satisfies $rc(G) \sim \max\{Z_1,\text{diam}(G)\}$ with high probability (whp). Here $Z_1$ is the number of vertices in $G$ whose degree equals 1 and the diameter of $G$ is asymptotically equal to $\frac{\log n}{\log\log n}$ whp. Finally, we prove that the rainbow connectivity $rc(G)$ of the random $r$-regular graph $G=G(n,r)$ whp satisfies $rc(G) =O(\log^{2\theta_r}{n})$ where $\theta_r=\frac{\log (r-1)}{\log (r-2)}$ when $r\geq 4$ and $rc(G) =O(\log^4n)$ whp when $r=3$.

scholarly journals Coverings, Heat Kernels and Spanning Trees

10.37236/1444 ◽  
1998 ◽  
Vol 6 (1) ◽  
Author(s):  
Fan Chung ◽  
S.-T. Yau
Keyword(s):  
Heat Kernel ◽  
Regular Graph ◽  
Spanning Trees ◽  
Constant Factor ◽  
Heat Kernels ◽  
Regular Graphs ◽  
Regular Tree

We consider a graph $G$ and a covering $\tilde{G}$ of $G$ and we study the relations of their eigenvalues and heat kernels. We evaluate the heat kernel for an infinite $k$-regular tree and we examine the heat kernels for general $k$-regular graphs. In particular, we show that a $k$-regular graph on $n$ vertices has at most $$ (1+o(1)) {{2\log n}\over {kn \log k}} \left( {{ (k-1)^{k-1}}\over {(k^2-2k)^{k/2-1}}}\right)^n $$ spanning trees, which is best possible within a constant factor.

scholarly journals Subgraphs of Randomk-Edge-Colouredk-Regular Graphs

2009 ◽  
Vol 18 (4) ◽  
pp. 533-549 ◽  
Author(s):  
PAULETTE LIEBY ◽  
BRENDAN D. McKAY ◽  
JEANETTE C. McLEOD ◽  
IAN M. WANLESS
Keyword(s):  
Regular Graph ◽  
Asymptotic Estimate ◽  
Forthcoming Paper ◽  
Perfect Matchings ◽  
Regular Graphs ◽  
Random Set ◽  
Expected Number ◽  
Edge Disjoint ◽  
Edge Colouring

LetG=G(n) be a randomly chosenk-edge-colouredk-regular graph with 2nvertices, wherek=k(n). Such a graph can be obtained from a random set ofkedge-disjoint perfect matchings ofK2n. Leth=h(n) be a graph withm=m(n) edges such thatm2+mk=o(n). Using a switching argument, we find an asymptotic estimate of the expected number of subgraphs ofGisomorphic toh. Isomorphisms may or may not respect the edge colouring, and other generalizations are also presented. Special attention is paid to matchings and cycles.The results in this paper are essential to a forthcoming paper of McLeod in which an asymptotic estimate for the number ofk-edge-colouredk-regular graphs fork=o(n5/6) is found.

Sign in / Sign up

Export Citation Format

Share Document