scholarly journals Another Antimagic Conjecture

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2071
Author(s):  
Rinovia Simanjuntak ◽  
Tamaro Nadeak ◽  
Fuad Yasin ◽  
Kristiana Wijaya ◽  
Nurdin Hinding ◽  
...  
Keyword(s):  
Connected Graph ◽  
Disjoint Union ◽  
Regular Graphs ◽  
Computational Results ◽  
Antimagic Labeling

An antimagic labeling of a graph G is a bijection f:E(G)→{1,…,|E(G)|} such that the weights w(x)=∑y∼xf(y) distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1990) is that every connected graph other than K2 admits an antimagic labeling. For a set of distances D, a D-antimagic labeling of a graph G is a bijection f:V(G)→{1,…,|V(G)|} such that the weightω(x)=∑y∈ND(x)f(y) is distinct for each vertex x, where ND(x)={y∈V(G)|d(x,y)∈D} is the D-neigbourhood set of a vertex x. If ND(x)=r, for every vertex x in G, a graph G is said to be (D,r)-regular. In this paper, we conjecture that a graph admits a D-antimagic labeling if and only if it does not contain two vertices having the same D-neighborhood set. We also provide evidence that the conjecture is true. We present computational results that, for D={1}, all graphs of order up to 8 concur with the conjecture. We prove that the set of (D,r)-regular D-antimagic graphs is closed under union. We provide examples of disjoint union of symmetric (D,r)-regular that are D-antimagic and examples of disjoint union of non-symmetric non-(D,r)-regular graphs that are D-antimagic. Furthermore, lastly, we show that it is possible to obtain a D-antimagic graph from a previously known distance antimagic graph.

scholarly journals A Proof of the Two-path Conjecture

10.37236/1665 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Herbert Fleischner ◽  
Robert R. Molina ◽  
Ken W. Smith ◽  
Douglas B. West
Keyword(s):  
Connected Graph ◽  
Disjoint Union ◽  
Hamiltonian Cycles ◽  
Regular Graphs ◽  
Edge Disjoint

Let $G$ be a connected graph that is the edge-disjoint union of two paths of length $n$, where $n\ge2$. Using a result of Thomason on decompositions of 4-regular graphs into pairs of Hamiltonian cycles, we prove that $G$ has a third path of length $n$.

scholarly journals Enumeration of the Edge Weights of Symmetrically Designed Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Muhammad Javaid ◽  
Hafiz Usman Afzal ◽  
Ebenezer Bonyah
Keyword(s):  
Disjoint Union ◽  
Antimagic Labeling ◽  
Pancyclic Graphs

The idea of super a , 0 -edge-antimagic labeling of graphs had been introduced by Enomoto et al. in the late nineties. This article addresses super a , 0 -edge-antimagic labeling of a biparametric family of pancyclic graphs. We also present the aforesaid labeling on the disjoint union of graphs comprising upon copies of C 4 and different trees. Several problems shall also be addressed in this article.

Super face d-antimagic labeling for disjoint union of toroidal fullerenes

2016 ◽  
Vol 55 (3) ◽  
pp. 849-863 ◽  
Author(s):  
Shahid Imran ◽  
Muhammad Hussain ◽  
Muhammad Kamran Siddiqui ◽  
Muhammad Numan
Keyword(s):  
Disjoint Union ◽  
Antimagic Labeling

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.

Construction of Antimagic Labeling for the Cartesian Product of Regular Graphs

2011 ◽  
Vol 5 (1) ◽  
pp. 81-87 ◽  
Author(s):  
Oudone Phanalasy ◽  
Mirka Miller ◽  
Costas S. Iliopoulos ◽  
Solon P. Pissis ◽  
Elaheh Vaezpour
Keyword(s):  
Cartesian Product ◽  
Regular Graphs ◽  
Antimagic Labeling

scholarly journals Antimagic Labeling of Regular Graphs

2015 ◽  
Vol 82 (4) ◽  
pp. 339-349 ◽  
Author(s):  
Feihuang Chang ◽  
Yu-Chang Liang ◽  
Zhishi Pan ◽  
Xuding Zhu
Keyword(s):  
Regular Graphs ◽  
Antimagic Labeling

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 THE VERTEX DISTANCE COMPLEMENT SPECTRUM OF SUBDIVISION VERTEX JOIN AND SUBDIVISION EDGE JOIN OF TWO REGULAR GRAPHS

2021 ◽  
Vol 7 (1) ◽  
pp. 102
Author(s):  
Ann Susa Thomas ◽  
Sunny Joseph Kalayathankal ◽  
Joseph Varghese Kureethara
Keyword(s):  
Connected Graph ◽  
Symmetric Matrix ◽  
Regular Graphs ◽  
Real Symmetric Matrix ◽  
Adjacency Spectrum ◽  
Vertex Set

The vertex distance complement (VDC) matrix \(\textit{C}\), of a connected graph  \(G\) with vertex set consisting of \(n\) vertices, is a real symmetric matrix \([c_{ij}]\) that takes the value \(n - d_{ij}\) where \(d_{ij}\) is the distance between the vertices \(v_i\) and \(v_j\) of \(G\) for \(i \neq j\) and 0 otherwise. The vertex distance complement spectrum of the subdivision vertex join, \(G_1 \dot{\bigvee} G_2\) and the subdivision edge join \(G_1 \underline{\bigvee} G_2\) of regular graphs \(G_1\) and \(G_2\)  in terms of the adjacency spectrum are determined in this paper.

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.

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