Coverings, Heat Kernels and Spanning Trees

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.

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On the Number of Spanning Trees in Random Regular Graphs

The Electronic Journal of Combinatorics ◽

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.

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Optimal Construction of Edge-Disjoint Paths in Random Regular Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548300004284 ◽

2000 ◽

Vol 9 (3) ◽

pp. 241-263 ◽

Cited By ~ 9

Author(s):

ALAN M. FRIEZE ◽

LEI ZHAO

Keyword(s):

Polynomial Time ◽

Regular Graph ◽

Randomized Algorithm ◽

Constant Factor ◽

Disjoint Paths ◽

Regular Graphs ◽

Edge Disjoint ◽

Random Regular Graph ◽

Arbitrary Graphs ◽

Optimal Construction

Given a graph G = (V, E) and a set of κ pairs of vertices in V, we are interested in finding, for each pair (ai, bi), a path connecting ai to bi such that the set of κ paths so found is edge-disjoint. (For arbitrary graphs the problem is [Nscr ][Pscr ]-complete, although it is in [Pscr ] if κ is fixed.)We present a polynomial time randomized algorithm for finding edge-disjoint paths in the random regular graph Gn,r, for sufficiently large r. (The graph is chosen first, then an adversary chooses the pairs of end-points.) We show that almost every Gn,r is such that all sets of κ = Ω(n/log n) pairs of vertices can be joined. This is within a constant factor of the optimum.

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On the Resistance Matrix of a Graph

The Electronic Journal of Combinatorics ◽

10.37236/5295 ◽

2016 ◽

Vol 23 (1) ◽

Cited By ~ 7

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 $i,j\in V(G)$.

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Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

Nagoya Mathematical Journal ◽

10.1017/s0027763000009958 ◽

2010 ◽

Vol 198 ◽

pp. 121-172

Author(s):

Gautam Chinta ◽

Jay Jorgenson ◽

Anders Karlsson

Keyword(s):

Heat Kernel ◽

Modular Forms ◽

Spanning Trees ◽

Zeta Functions ◽

Constant Term ◽

Finite Graph ◽

Heat Kernels ◽

Careful Study ◽

Trigonometric Functions ◽

Combinatorial Laplacian

AbstractBy a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with the generating set given by choosing a generator for each cyclic factor. In this article we examine the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First, we show that the sequence of heat kernels corresponding to the degenerating family converges, after rescaling, to the heat kernel on an associated real torus. We then establish an asymptotic expansion, in the degeneration parameter, of the determinant of the combinatorial Laplacian. The zeta-regularized determinant of the Laplacian of the limiting real torus appears as the constant term in this expansion. On the other hand, using a classical theorem by Kirchhoff, the determinant of the combinatorial Laplacian of a finite graph divided by the number of vertices equals the number of spanning trees, called thecomplexity, of the graph. As a result, we establish a precise connection between the complexity of the Cayley graphs of finite abelian groups and heights of real tori. It is also known that spectral determinants on discrete tori can be expressed using trigonometric functions and that spectral determinants on real tori can be expressed using modular forms on general linear groups. Another interpretation of our analysis is thus to establish a link between limiting values of certain products of trigonometric functions and modular forms. The heat kernel analysis which we employ uses a careful study ofI-Bessel functions. Our methods extend to prove the asymptotic behavior of other spectral invariants through degeneration, such as special values of spectral zeta functions and Epstein-Hurwitz–type zeta functions.

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Coverings, Laplacians, and Heat Kernels of Directed Graphs

The Electronic Journal of Combinatorics ◽

10.37236/120 ◽

2009 ◽

Vol 16 (1) ◽

Cited By ~ 3

Author(s):

Clara E. Brasseur ◽

Ryan E. Grady ◽

Stratos Prassidis

Keyword(s):

Random Walk ◽

Heat Kernel ◽

Directed Graphs ◽

Heat Kernels ◽

Total Space ◽

Regular Tree ◽

Combinatorial Laplacian

Combinatorial covers of graphs were defined by Chung and Yau. Their main feature is that the spectra of the Combinatorial Laplacian of the base and the total space are related. We extend their definition to directed graphs. As an application, we compute the spectrum of the Combinatorial Laplacian of the homesick random walk $RW_{\mu}$ on the line. Using this calculation, we show that the heat kernel on the weighted line can be computed from the heat kernel of '$(1 + 1/\mu)$-regular' tree.

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Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

Nagoya Mathematical Journal ◽

10.1215/00277630-2009-009 ◽

2010 ◽

Vol 198 ◽

pp. 121-172 ◽

Cited By ~ 11

Author(s):

Gautam Chinta ◽

Jay Jorgenson ◽

Anders Karlsson

Keyword(s):

Heat Kernel ◽

Modular Forms ◽

Spanning Trees ◽

Zeta Functions ◽

Constant Term ◽

Finite Graph ◽

Heat Kernels ◽

Careful Study ◽

Trigonometric Functions ◽

Combinatorial Laplacian

AbstractBy a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with the generating set given by choosing a generator for each cyclic factor. In this article we examine the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First, we show that the sequence of heat kernels corresponding to the degenerating family converges, after rescaling, to the heat kernel on an associated real torus. We then establish an asymptotic expansion, in the degeneration parameter, of the determinant of the combinatorial Laplacian. The zeta-regularized determinant of the Laplacian of the limiting real torus appears as the constant term in this expansion. On the other hand, using a classical theorem by Kirchhoff, the determinant of the combinatorial Laplacian of a finite graph divided by the number of vertices equals the number of spanning trees, called the complexity, of the graph. As a result, we establish a precise connection between the complexity of the Cayley graphs of finite abelian groups and heights of real tori. It is also known that spectral determinants on discrete tori can be expressed using trigonometric functions and that spectral determinants on real tori can be expressed using modular forms on general linear groups. Another interpretation of our analysis is thus to establish a link between limiting values of certain products of trigonometric functions and modular forms. The heat kernel analysis which we employ uses a careful study of I-Bessel functions. Our methods extend to prove the asymptotic behavior of other spectral invariants through degeneration, such as special values of spectral zeta functions and Epstein-Hurwitz–type zeta functions.

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Cycle partitions of regular graphs

Combinatorics Probability Computing ◽

10.1017/s0963548320000553 ◽

2020 ◽

pp. 1-24

Author(s):

Vytautas Gruslys ◽

Shoham Letzter

Keyword(s):

Regular Graph ◽

Bipartite Graphs ◽

Disjoint Paths ◽

Regular Graphs ◽

Simple Construction ◽

Vertex Set ◽

Almost All ◽

Vertex Disjoint

Abstract Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

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

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972715000027 ◽

2015 ◽

Vol 91 (3) ◽

pp. 353-367 ◽

Cited By ~ 38

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 $q(G)$).

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The maximum number of vertices of primitive regular graphs of orders 2, 3, 4 with exponent 2

PRIKLADNAYa DISKRETNAYa MATEMATIKA ◽

10.17223/20710410/52/6 ◽

2021 ◽

pp. 97-104

Author(s):

M. B. Abrosimov ◽

◽

S. V. Kostin ◽

I. V. Los ◽

◽

...

Keyword(s):

Regular Graph ◽

Regular Graphs ◽

Minimal Regular Graphs of Girths Eight and Twelve

Canadian Journal of Mathematics ◽

10.4153/cjm-1966-109-8 ◽

1966 ◽

Vol 18 ◽

pp. 1091-1094 ◽

Cited By ~ 104

Author(s):

Clark T. Benson

Keyword(s):

Group Theory ◽

Lower Bound ◽

Regular Graph ◽

Regular Graphs ◽

Image Position

In (3) Tutte showed that the order of a regular graph of degree d and even girth g > 4 is greater than or equal toHere the girth of a graph is the length of the shortest circuit. It was shown in (2) that this lower bound cannot be attained for regular graphs of degree > 2 for g ≠ 6, 8, or 12. When this lower bound is attained, the graph is called minimal. In a group-theoretic setting a similar situation arose and it was noticed by Gleason that minimal regular graphs of girth 12 could be constructed from certain groups. Here we construct these graphs making only incidental use of group theory. Also we give what is believed to be an easier construction of minimal regular graphs of girth 8 than is given in (2). These results are contained in the following two theorems.

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