scholarly journals The Classification of $2$-Extendable Edge-Regular Graphs with Diameter $2$

10.37236/8073 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Klavdija Kutnar ◽  
Dragan Marušič ◽  
Štefko Miklavič ◽  
Primož Šparl
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Perfect Matching ◽  
Negative Integer ◽  
Regular Graphs ◽  
Even Order

Let $\ell$ denote a non-negative integer. A connected graph $\Gamma$ of even order at least $2\ell+2$ is $\ell$-extendable if it contains a matching of size $\ell$ and if every such matching is contained in a perfect matching of $\Gamma$. A connected regular graph $\Gamma$ is edge-regular, if there exists an integer $\lambda$ such that every pair of adjacent vertices of $\Gamma$ have exactly $\lambda$  common neighbours. In this paper we classify $2$-extendable edge-regular graphs of even order and diameter $2$.

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 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 0-Sum and 1-Sum Flows in Regular Graphs

10.37236/4986 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
S. Akbari ◽  
M. Kano ◽  
S. Zare
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Natural Numbers ◽  
Real Numbers ◽  
Even Order

Let $G$ be a graph. Assume that $l$ and $k$ are two natural numbers. An $l$-sum flow on a graph $G$ is an assignment of non-zero real numbers to the edges of $G$ such that for every vertex $v$ of $G$ the sum of values of all edges incidence with $v$ equals $l$. An $l$-sum $k$-flow is an $l$-sum flow with values from the set $\{\pm 1,\ldots ,\pm(k-1)\}$. Recently, it was proved that for every $r, r\geq 3$, $r\neq 5$, every $r$-regular graph admits a $0$-sum $5$-flow. In this paper we settle a conjecture by showing that every $5$-regular graph admits a $0$-sum $5$-flow. Moreover, we prove that every $r$-regular graph of even order admits a $1$-sum $5$-flow.

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.

On the Largest Distance (Signless Laplacian) Eigenvalue of Non-transmission-regular Graphs

2018 ◽  
Vol 34 ◽  
pp. 459-471 ◽  
Author(s):  
Shuting Liu ◽  
Jinlong Shu ◽  
Jie Xue
Keyword(s):  
Spectral Radius ◽  
Regular Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Wiener Index ◽  
Regular Graphs ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Sum Of Distances

Let $G=(V(G),E(G))$ be a $k$-connected graph with $n$ vertices and $m$ edges. Let $D(G)$ be the distance matrix of $G$. Suppose $\lambda_1(D)\geq \cdots \geq \lambda_n(D)$ are the $D$-eigenvalues of $G$. The transmission of $v_i \in V(G)$, denoted by $Tr_G(v_i)$ is defined to be the sum of distances from $v_i$ to all other vertices of $G$, i.e., the row sum $D_{i}(G)$ of $D(G)$ indexed by vertex $v_i$ and suppose that $D_1(G)\geq \cdots \geq D_n(G)$. The $Wiener~ index$ of $G$ denoted by $W(G)$ is given by $W(G)=\frac{1}{2}\sum_{i=1}^{n}D_i(G)$. Let $Tr(G)$ be the $n\times n$ diagonal matrix with its $(i,i)$-entry equal to $TrG(v_i)$. The distance signless Laplacian matrix of $G$ is defined as $D^Q(G)=Tr(G)+D(G)$ and its spectral radius is denoted by $\rho_1(D^Q(G))$ or $\rho_1$. A connected graph $G$ is said to be $t$-transmission-regular if $Tr_G(v_i) =t$ for every vertex $v_i\in V(G)$, otherwise, non-transmission-regular. In this paper, we respectively estimate $D_1(G)-\lambda_1(G)$ and $2D_1(G)-\rho_1(G)$ for a $k$-connected non-transmission-regular graph in different ways and compare these obtained results. And we conjecture that $D_1(G)-\lambda_1(G)>\frac{1}{n+1}$. Moreover, we show that the conjecture is valid for trees.

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 A Relation between D-Index and Wiener Index for r-Regular Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Ahmed Mohammed Ali ◽  
Asmaa Salah Aziz
Keyword(s):  
General Formula ◽  
Regular Graph ◽  
Connected Graph ◽  
Wiener Index ◽  
Regular Graphs ◽  
Irregular Graphs

For any two distinct vertices u and  v in a connected graph G, let lPu,v=lP be the length of u−v path P and the D–distance between u and v of G is defined as: dDu,v=minplP+∑∀y∈VPdeg y, where the minimum is taken over all u−v paths P and the sum is taken over all vertices of u−v path P. The D-index of G is defined as WDG=1/2∑∀v,u∈VGdDu,v. In this paper, we found a general formula that links the Wiener index with D-index of a regular graph G. Moreover, we obtained different formulas of many special irregular graphs.

Pentavalent One-regular Graphs of Square-free Order

2010 ◽  
Vol 17 (03) ◽  
pp. 515-524 ◽  
Author(s):  
Yantao Li ◽  
Yan-Quan Feng
Keyword(s):  
Automorphism Group ◽  
Regular Graph ◽  
Cayley Graphs ◽  
Regular Graphs ◽  
Dihedral Groups ◽  
Free Order

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. Let n be a square-free integer. It is shown in this paper that a pentavalent one-regular graph of order n exists if and only if n = 2 · 5tp1p2 … ps ≥ 62, where t ≤ 1, s ≥ 1, and pi's are distinct primes such that 5|(pi-1). For such an integer n, there are exactly 4s-1 non-isomorphic pentavalent one-regular graphs of order n, which are Cayley graphs on dihedral groups constructed by Kwak et al. This work is a continuation of the classification of cubic one-regular graphs of order twice a square-free integer given by Zhou and Feng.

scholarly journals Extending a perfect matching to a Hamiltonian cycle

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Adel Alahmadi ◽  
Robert E. L. Aldred ◽  
Ahmad Alkenani ◽  
Rola Hijazi ◽  
P. Solé ◽  
...  
Keyword(s):  
Graph Theory ◽  
Regular Graph ◽  
Complete Graph ◽  
Hamiltonian Cycle ◽  
Perfect Matching ◽  
Regular Graphs ◽  
Spanning Subgraph ◽  
International Audience ◽  
Matching Extendability

Graph Theory International audience Ruskey and Savage conjectured that in the d-dimensional hypercube, every matching M can be extended to a Hamiltonian cycle. Fink verified this for every perfect matching M, remarkably even if M contains external edges. We prove that this property also holds for sparse spanning regular subgraphs of the cubes: for every d ≥7 and every k, where 7 ≤k ≤d, the d-dimensional hypercube contains a k-regular spanning subgraph such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle. We do not know if this result can be extended to k=4,5,6. It cannot be extended to k=3. Indeed, there are only three 3-regular graphs such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle, namely the complete graph on 4 vertices, the complete bipartite 3-regular graph on 6 vertices and the 3-cube on 8 vertices. Also, we do not know if there are graphs of girth at least 5 with this matching-extendability property.

scholarly journals Pan-Factorial Property in Regular Graphs

10.37236/1990 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
M. Kano ◽  
Qinglin Yu
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
K Factor ◽  
Even Order

Among other results, we show that if for any given edge $e$ of an $r$-regular graph $G$ of even order, $G$ has a 1-factor containing $e$, then $G$ has a $k$-factor containing $e$ and another one avoiding $e$ for all $k$, $1 \leq k \leq r-1$.

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