scholarly journals Improved Lower Bounds for the Orders of Even Girth Cages

10.37236/6015 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Tatiana Baginová Jajcayová ◽  
Slobodan Filipovski ◽  
Robert Jajcay
Keyword(s):  
Lower Bound ◽  
Structural Property ◽  
Lower Bounds ◽  
Regular Graph ◽  
Regular Graphs ◽  
The Difference

The well-known Moore bound $M(k,g)$ serves as a universal lower bound for the order of $k$-regular graphs of girth $g$. The excess $e$ of a $k$-regular graph $G$ of girth $g$ and order $n$ is the difference between its order $n$ and the corresponding Moore bound, $e=n - M(k,g) $. We find infinite families of parameters $(k,g)$, $g$ even, for which we show that the excess of any $k$-regular graph of girth $g$ is larger than $4$. This yields new improved lower bounds on the order of $k$-regular graphs of girth $g$ of smallest possible order; the so-called $(k,g)$-cages. We also show that the excess of the smallest $k$-regular graphs of girth $g$ can be arbitrarily large for a restricted family of $(k,g)$-graphs satisfying a very natural additional structural property.

Minimal Regular Graphs of Girths Eight and Twelve

1966 ◽  
Vol 18 ◽  
pp. 1091-1094 ◽  
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.

On the Non-Existence of a Type of Regular Graphs of Girth 5

1967 ◽  
Vol 19 ◽  
pp. 644-648 ◽  
Author(s):  
William G. Brown
Keyword(s):  
Lower Bound ◽  
Regular Graph ◽  
Regular Graphs ◽  
Minimal Graph ◽  
Image Position

ƒ(k, 5) is defined to be the smallest integer n for which there exists a regular graph of valency k and girth 5, having n vertices. In (3) it was shown that1.1Hoffman and Singleton proved in (4) that equality holds in the lower bound of (1.1) only for k = 2, 3, 7, and possibly 57. Robertson showed in (6) that ƒ(4, 5) = 19 and constructed the unique minimal graph.

scholarly journals Counting Proper Colourings in 4-Regular Graphs via the Potts Model

10.37236/7743 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Ewan Davies
Keyword(s):  
Partition Function ◽  
Internal Energy ◽  
Lower Bounds ◽  
Regular Graph ◽  
Potts Model ◽  
Temperature Limit ◽  
Upper And Lower Bounds ◽  
Regular Graphs ◽  
Zero Temperature ◽  
First Case

We give tight upper and lower bounds on the internal energy per particle in the antiferromagnetic $q$-state Potts model on $4$-regular graphs, for $q\ge 5$. This proves the first case of a conjecture of the author, Perkins, Jenssen and Roberts, and implies tight bounds on the antiferromagnetic Potts partition function. The zero-temperature limit gives upper and lower bounds on the number of proper $q$-colourings of $4$-regular graphs, which almost proves the case $d=4$ of a conjecture of Galvin and Tetali. For any $q \ge 5$ we prove that the number of proper $q$-colourings of a $4$-regular graph is maximised by a union of $K_{4,4}$'s.  

scholarly journals Induced Forests in Regular Graphs with Large Girth

2008 ◽  
Vol 17 (3) ◽  
pp. 389-410 ◽  
Author(s):  
CARLOS HOPPEN ◽  
NICHOLAS WORMALD
Keyword(s):  
Lower Bound ◽  
Regular Graph ◽  
Randomized Algorithm ◽  
Alternative Proof ◽  
Regular Graphs ◽  
Induced Subgraph ◽  
Large Girth

An induced forest of a graph G is an acyclic induced subgraph of G. The present paper is devoted to the analysis of a simple randomized algorithm that grows an induced forest in a regular graph. The expected size of the forest it outputs provides a lower bound on the maximum number of vertices in an induced forest of G. When the girth is large and the degree is at least 4, our bound coincides with the best bound known to hold asymptotically almost surely for random regular graphs. This results in an alternative proof for the random case.

scholarly journals Difference betwwen the maximum degree and the index of a graph

2019 ◽  
Vol 54 (4) ◽  
pp. 434-440
Author(s):  
V. I. Benediktovich
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Lower Boundary ◽  
Arbitrary Graph ◽  
Analogous Statement ◽  
Graph Classes ◽  
The Difference ◽  
A Chain

An algebraic parameter of a graph – a difference between its maximum degree and its spectral radius is considered in this paper. It is well known that this graph parameter is always nonnegative and represents some measure of deviation of a graph from its regularity. In the last two decades, many papers have been devoted to the study of this parameter. In particular, its lower bound depending on the graph order and diameter was obtained in 2007 by mathematician S. M. Cioabă. In 2017 when studying the upper and the lower bounds of this parameter, M. R. Oboudi made a conjecture that the lower bound of a given parameter for an arbitrary graph is the difference between a maximum degree and a spectral radius of a chain. This is very similar to the analogous statement for the spectral radius of an arbitrary graph whose lower boundary is also the spectral radius of a chain. In this paper, the above conjecture is confirmed for some graph classes.

scholarly journals On the Dα spectral radius of strongly connected digraphs

Filomat ◽  
2021 ◽  
Vol 35 (4) ◽  
pp. 1289-1304
Author(s):  
Weige Xi
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Lower Bounds ◽  
Distance Matrix ◽  
Upper And Lower Bounds ◽  
Strongly Connected Digraph ◽  
Connected Digraph ◽  
The Matrix ◽  
Strongly Connected ◽  
The Difference

Let G be a strongly connected digraph with distance matrix D(G) and let Tr(G) be the diagonal matrix with vertex transmissions of G. For any real ? ? [0, 1], define the matrix D?(G) as D?(G) = ?Tr(G) + (1-?)D(G). The D? spectral radius of G is the spectral radius of D?(G). In this paper, we first give some upper and lower bounds for the D? spectral radius of G and characterize the extremal digraphs. Moreover, for digraphs that are not transmission regular, we give a lower bound on the difference between the maximum vertex transmission and the D? spectral radius. Finally, we obtain the D? eigenvalues of the join of certain regular digraphs.

scholarly journals Lower Bounds for the Cop Number when the Robber is Fast

2011 ◽  
Vol 20 (4) ◽  
pp. 617-621 ◽  
Author(s):  
ABBAS MEHRABIAN
Keyword(s):  
Graph Theory ◽  
Lower Bound ◽  
Lower Bounds ◽  
Regular Graph ◽  
Upper Bound ◽  
Upper Bounds ◽  
Cops And Robbers ◽  
Vertex Graph

We consider a variant of the Cops and Robbers game where the robber can movetedges at a time, and show that in this variant, the cop number of ad-regular graph with girth larger than 2t+2 is Ω(dt). By the known upper bounds on the order of cages, this implies that the cop number of a connectedn-vertex graph can be as large as Ω(n2/3) ift≥ 2, and Ω(n4/5) ift≥ 4. This improves the Ω($n^{\frac{t-3}{t-2}}$) lower bound of Frieze, Krivelevich and Loh (Variations on cops and robbers,J. Graph Theory, to appear) when 2 ≤t≤ 6. We also conjecture a general upper boundO(nt/t+1) for the cop number in this variant, generalizing Meyniel's conjecture.

scholarly journals LANG'S CONJECTURE AND SHARP HEIGHT ESTIMATES FOR THE ELLIPTIC CURVES y2 = x3 + ax

2013 ◽  
Vol 09 (05) ◽  
pp. 1141-1170 ◽  
Author(s):  
PAUL VOUTIER ◽  
MINORU YABUTA
Keyword(s):  
Lower Bound ◽  
Elliptic Curves ◽  
Lower Bounds ◽  
Rational Points ◽  
Upper And Lower Bounds ◽  
Canonical Height ◽  
The Difference ◽  
Lang's Conjecture ◽  
Lang’S Conjecture ◽  
Logarithmic Height

For elliptic curves given by the equation Ea : y2 = x3 + ax, we establish the best-possible version of Lang's conjecture on the lower bound for the canonical height of non-torsion rational points along with best-possible upper and lower bounds for the difference between the canonical and logarithmic height.

scholarly journals Regular Factors of Regular Graphs from Eigenvalues

10.37236/431 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Hongliang Lu
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Regular Graph ◽  
Maximum Degree ◽  
Upper Bounds ◽  
Regular Graphs ◽  
Largest Eigenvalue ◽  
The Third ◽  
K Factor ◽  
Second Largest Eigenvalue

Let $r$ and $m$ be two integers such that $r\geq m$. Let $H$ be a graph with order $|H|$, size $e$ and maximum degree $r$ such that $2e\geq |H|r-m$. We find a best lower bound on spectral radius of graph $H$ in terms of $m$ and $r$. Let $G$ be a connected $r$-regular graph of order $|G|$ and $ k < r$ be an integer. Using the previous results, we find some best upper bounds (in terms of $r$ and $k$) on the third largest eigenvalue that is sufficient to guarantee that $G$ has a $k$-factor when $k|G|$ is even. Moreover, we find a best bound on the second largest eigenvalue that is sufficient to guarantee that $G$ is $k$-critical when $k|G|$ is odd. Our results extend the work of Cioabă, Gregory and Haemers [J. Combin. Theory Ser. B, 1999] who obtained such results for 1-factors.

A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Ramsey Numbers

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.

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