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.

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

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 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 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 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.

scholarly journals On $k$-Walk-Regular Graphs

10.37236/136 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
C. Dalfó ◽  
M. A. Fiol ◽  
E. Garriga
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
The Other ◽  
Regular Graphs ◽  
Local Spectrum ◽  
Number Of Cycles

Considering a connected graph $G$ with diameter $D$, we say that it is $k$-walk-regular, for a given integer $k$ $(0\leq k \leq D)$, if the number of walks of length $\ell$ between any pair of vertices only depends on the distance between them, provided that this distance does not exceed $k$. Thus, for $k=0$, this definition coincides with that of walk-regular graph, where the number of cycles of length $\ell$ rooted at a given vertex is a constant through all the graph. In the other extreme, for $k=D$, we get one of the possible definitions for a graph to be distance-regular. In this paper we show some algebraic characterizations of $k$-walk-regularity, which are based on the so-called local spectrum and predistance polynomials of $G$.

Some Spectral Characterizations of Strongly Distance-Regular Graphs

2001 ◽  
Vol 10 (2) ◽  
pp. 127-135 ◽  
Author(s):  
M. A. FIOL
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Distance Regular Graph ◽  
Strongly Regular ◽  
Distance Regular Graphs ◽  
Spectral Characterizations

A graph Γ with diameter d is strongly distance-regular if Γ is distance-regular and its distance-d graph Γd is strongly regular. Some known examples of such graphs are the connected strongly regular graphs, with distance-d graph Γd = Γ (the complement of Γ), and the antipodal distance-regular graphs. Here we study some spectral conditions for a (regular or distance-regular) graph to be strongly distance-regular. In particular, for the case d = 3 the following characterization is proved. A regular (connected) graph Γ, with distinct eigenvalues λ0 > λ1 > λ2 > λ3, is strongly distance-regular if and only if λ2 = −1, and Γ3 is k-regular with degree k satisfying an expression which depends only on the order and the different eigenvalues of Γ.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

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