scholarly journals Fractional Decompositions and the Smallest-eigenvalue Separation

10.37236/8833 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Fiachra Knox ◽  
Bojan Mohar
Keyword(s):  
Recent Result ◽  
Regular Graph ◽  
Short Proof ◽  
New Method ◽  
Regular Graphs ◽  
Smallest Eigenvalue ◽  
Distance Regular Graphs

A new method is introduced for bounding the separation between the value of $-k$ and the smallest eigenvalue of a non-bipartite $k$-regular graph. The method is based on fractional decompositions of graphs. As a consequence we obtain a very short proof of a generalization and strengthening of a recent result of Qiao, Jing, and Koolen [Electronic J. Combin. 26(2) (2019), #P2.41] about the smallest eigenvalue of non-bipartite distance-regular graphs.

scholarly journals Non-Bipartite Distance-Regular Graphs with a Small Smallest Eigenvalue

10.37236/8361 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Zhi Qiao ◽  
Yifan Jing ◽  
Jack Koolen
Keyword(s):  
Fixed Number ◽  
Regular Graphs ◽  
Fixed Integer ◽  
Smallest Eigenvalue ◽  
Distance Regular Graphs

In 2017, Qiao and Koolen showed that for any fixed integer $D\geqslant 3$, there are only finitely many such graphs with $\theta_{\min}\leqslant -\alpha k$, where $0<\alpha<1$ is any fixed number. In this paper, we will study non-bipartite distance-regular graphs with relatively small $\theta_{\min}$ compared with $k$. In particular, we will show that if $\theta_{\min}$ is relatively close to $-k$, then the odd girth $g$ must be large. Also we will classify the non-bipartite distance-regular graphs with $\theta_{\min} \leqslant -\frac{D-1}{D}k$ for $D =4,5$.

Merging the first and third classes in bipartite distance-regular graphs

2019 ◽  
Vol 12 (07) ◽  
pp. 2050009
Author(s):  
Siwaporn Mamart ◽  
Chalermpong Worawannotai
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Original Graph ◽  
Distance Regular Graphs

Merging the first and third classes in a connected graph is the operation of adding edges between all vertices at distance 3 in the original graph while keeping the original edges. We determine when merging the first and third classes in a bipartite distance-regular graph produces a distance-regular graph.

scholarly journals Thin Distance-Regular Graphs with Classical Parameters $(D,q,q, \frac{q^{t}-1}{q-1}-1)$ with $t> D$ are the Grassmann Graphs

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Ying Ying Tan ◽  
Xiaoye Liang ◽  
Jack Koolen
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Intersection Numbers ◽  
Survey Paper ◽  
Grassmann Graph ◽  
Distance Regular Graphs ◽  
Classical Parameters

In the survey paper by Van Dam, Koolen and Tanaka (2016), they asked to classify the thin $Q$-polynomial distance-regular graphs. In this paper, we show that a thin distance-regular graph with the same intersection numbers as a Grassmann graph $J_q(n, D)~ (n \geqslant 2D)$ is the Grassmann graph if $D$ is large enough.

scholarly journals Constrained Edge-Splitting Problems

1999 ◽  
Vol 6 (37) ◽  
Author(s):  
Tibor Jordán
Keyword(s):  
Recent Result ◽  
Structural Properties ◽  
Polynomial Algorithm ◽  
Short Proof ◽  
New Method ◽  
Minimum Size ◽  
Intersection Theorem ◽  
Edge Connectivity

<p>Splitting off two edges su, sv in a graph G means deleting su, sv and<br />adding a new edge uv. Let G = (V +s,E) be k-edge-connected in V<br />(k >= 2) and let d(s) be even. Lov´asz proved that the edges incident to s<br />can be split off in pairs in such a way that the resulting graph on vertex<br />set V is k-edge-connected. In this paper we investigate the existence of<br />such complete splitting sequences when the set of split edges has to meet<br />additional requirements. We prove structural properties of the set of those<br />pairs u, v of neighbours of s for which splitting off su, sv destroys k-edge-connectivity. This leads to a new method for solving problems of this type.</p><p>By applying this method we obtain a short proof for a recent result of<br />Nagamochi and Eades on planarity-preserving complete splitting sequences and prove the following new results: let G and H be two graphs on the same set V + s of vertices and suppose that their sets of edges incident to s coincide. Let G (H) be k-edge-connected (l-edge-connected, respectively) in V and let d(s) be even. Then there exists a pair su, sv which can be split off in both graphs preserving k-edge-connectivity (l-edge-connectivity, resp.) in V , provided d(s) >= 6. If k and l are both even then such a pair always exists. Using these edge-splitting results and the polymatroid intersection theorem we give a polynomial algorithm for the problem of simultaneously augmenting the edge-connectivity of two graphs by adding a (common) set of new edges of (almost) minimum size.</p>

scholarly journals A Short Proof of the Odd-Girth Theorem

10.37236/2289 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Edwin R. Van Dam ◽  
Miquel Angel Fiol
Keyword(s):  
Connected Graph ◽  
Short Proof ◽  
Direct Proof ◽  
Regular Graphs ◽  
Distance Regular Graphs

Recently, it has been shown that a connected graph $\Gamma$ with $d+1$ distinct eigenvalues and odd-girth $2d+1$ is distance-regular. The proof of this result was based on the spectral excess theorem. In this note we present an alternative and more direct proof which does not rely on the spectral excess theorem, but on a known characterization of distance regular graphs in terms of the predistance polynomial of degree $d$.

scholarly journals Irregularity Strength of Regular Graphs

10.37236/806 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Jakub Przybyło
Keyword(s):  
Positive Integer ◽  
Recent Result ◽  
Regular Graph ◽  
Simple Graph ◽  
Regular Graphs ◽  
Isolated Vertex ◽  
Irregularity Strength

Let $G$ be a simple graph with no isolated edges and at most one isolated vertex. For a positive integer $w$, a $w$-weighting of $G$ is a map $f:E(G)\rightarrow \{1,2,\ldots,w\}$. An irregularity strength of $G$, $s(G)$, is the smallest $w$ such that there is a $w$-weighting of $G$ for which $\sum_{e:u\in e}f(e)\neq\sum_{e:v\in e}f(e)$ for all pairs of different vertices $u,v\in V(G)$. A conjecture by Faudree and Lehel says that there is a constant $c$ such that $s(G)\le{n\over d}+c$ for each $d$-regular graph $G$, $d\ge 2$. We show that $s(G) < 16{n\over d}+6$. Consequently, we improve the results by Frieze, Gould, Karoński and Pfender (in some cases by a $\log n$ factor) in this area, as well as the recent result by Cuckler and Lazebnik.

scholarly journals On Bipartite $Q$-Polynomial Distance-Regular Graphs with Diameter 9, 10, or 11

10.37236/7347 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Štefko Miklavič
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Intersection Numbers ◽  
Distance Regular Graphs

Let $\Gamma$ denote a bipartite distance-regular graph with diameter $D$. In [Caughman (2004)], Caughman showed that if $D \ge 12$, then $\Gamma$ is $Q$-polynomial if and only if one of the following (i)-(iv) holds: (i) $\Gamma$ is the ordinary $2D$-cycle, (ii) $\Gamma$ is the Hamming cube $H(D,2)$, (iii) $\Gamma$ is the antipodal quotient of the Hamming cube $H(2D,2)$, (iv) the intersection numbers of $\Gamma$ satisfy $c_i = (q^i - 1)/(q-1)$, $b_i = (q^D-q^i)/(q-1)$ $(0 \le i \le D)$, where $q$ is an integer at least $2$. In this paper we show that the above result is true also for bipartite distance-regular graphs with $D \in \{9,10,11\}$.

scholarly journals The Spectral Excess Theorem for Distance-Regular Graphs: A Global (Over)view

10.37236/853 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Edwin R. Van Dam
Keyword(s):  
Regular Graph ◽  
Current Paper ◽  
Regular Graphs ◽  
Global Approach ◽  
Local Approach ◽  
Extremal Distance ◽  
Distance Regular Graphs

Distance-regularity of a graph is in general not determined by the spectrum of the graph. The spectral excess theorem states that a connected regular graph is distance-regular if for every vertex, the number of vertices at extremal distance (the excess) equals some given expression in terms of the spectrum of the graph. This result was proved by Fiol and Garriga [From local adjacency polynomials to locally pseudo-distance-regular graphs, J. Combinatorial Th. B 71 (1997), 162-183] using a local approach. This approach has the advantage that more general results can be proven, but the disadvantage that it is quite technical. The aim of the current paper is to give a less technical proof by taking a global approach.

scholarly journals Using Symbolic Computation to Prove Nonexistence of Distance-Regular Graphs

10.37236/7763 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Janoš Vidali
Keyword(s):  
Symbolic Computation ◽  
Computer Algebra ◽  
Regular Graph ◽  
Computer Algebra System ◽  
Intersection Array ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Triple Intersection ◽  
Distance Regular Graphs ◽  
Krein Condition

A package for the Sage computer algebra system is developed for checking feasibility of a given intersection array for a distance-regular graph. We use this tool to show that there is no distance-regular graph with intersection array$$\{(2r+1)(4r+1)(4t-1), 8r(4rt-r+2t), (r+t)(4r+1); 1, (r+t)(4r+1), 4r(2r+1)(4t-1)\}  (r, t \geq 1),$$$\{135,\! 128,\! 16; 1,\! 16,\! 120\}$, $\{234,\! 165,\! 12; 1,\! 30,\! 198\}$ or $\{55,\! 54,\! 50,\! 35,\! 10; 1,\! 5,\! 20,\! 45,\! 55\}$. In all cases, the proofs rely on equality in the Krein condition, from which triple intersection numbers are determined. Further combinatorial arguments are then used to derive nonexistence. 

scholarly journals On Bipartite $Q$-Polynomial Distance-Regular Graphs with $c_2 \le 2$

10.37236/4556 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Stefko Miklavic ◽  
Safet Penjic
Keyword(s):  
Regular Graph ◽  
Intersection Number ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Distance Regular Graphs

Let $\Gamma$ denote a bipartite $Q$-polynomial distance-regular graph with diameter $D \ge 4$, valency $k \ge 3$ and intersection number $c_2 \le 2$. We show that $\Gamma$ is either the $D$-dimensional hypercube, or the antipodal quotient of the $2D$-dimensional hypercube, or $D=5$.

Sign in / Sign up

Export Citation Format

Share Document