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 An Eigenvalue Characterization of Antipodal Distance-Regular Graphs

10.37236/1315 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
M. A. Fiol
Keyword(s):  
Regular Graph ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Graph ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Distance Regular Graphs

Let $G$ be a regular (connected) graph with $n$ vertices and $d+1$ distinct eigenvalues. As a main result, it is shown that $G$ is an $r$-antipodal distance-regular graph if and only if the distance graph $G_d$ is constituted by disjoint copies of the complete graph $K_r$, with $r$ satisfying an expression in terms of $n$ and the distinct eigenvalues.

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 The Spectral Excess Theorem for Distance-Biregular Graphs.

10.37236/3305 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Miquel Àngel Fiol
Keyword(s):  
Bipartite Graph ◽  
Connected Graph ◽  
Intersection Array ◽  
Stable Set ◽  
Regular Graphs ◽  
Distance Regular Graphs

The spectral excess theorem for distance-regular graphs states that a regular (connected) graph is distance-regular if and only if its spectral-excess equals its average excess. A bipartite graph $\Gamma$ is distance-biregular when it is distance-regular around each vertex and the intersection array only depends on the stable set such a vertex belongs to. In this note we derive a new version of the spectral excess theorem for bipartite distance-biregular graphs.

Characterization of certain distance-regular graphs by forbidden subgraphs

2007 ◽  
Vol 75 (3) ◽  
pp. 420-423 ◽  
Author(s):  
V. V. Kabanov ◽  
A. A. Makhnev ◽  
D. V. Paduchikh
Keyword(s):  
Regular Graphs ◽  
Forbidden Subgraphs ◽  
Distance Regular Graphs

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.

Sign in / Sign up

Export Citation Format

Share Document