scholarly journals Repetition Number of Graphs

10.37236/96 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Yair Caro ◽  
Douglas B. West
Keyword(s):  
Minimum Degree ◽  
Line Graph ◽  
Outerplanar Graph ◽  
Line Graphs ◽  
Outerplanar Graphs ◽  
Free Graph ◽  
Maximal Outerplanar Graphs ◽  
Vertex Graph ◽  
Repetition Number ◽  
Maximum Multiplicity

Every $n$-vertex graph has two vertices with the same degree (if $n\ge2$). In general, let rep$(G)$ be the maximum multiplicity of a vertex degree in $G$. An easy counting argument yields rep$(G)\ge n/(2d-2s+1)$, where $d$ is the average degree and $s$ is the minimum degree of $G$. Equality can hold when $2d$ is an integer, and the bound is approximately sharp in general, even when $G$ is restricted to be a tree, maximal outerplanar graph, planar triangulation, or claw-free graph. Among large claw-free graphs, repetition number $2$ is achievable, but if $G$ is an $n$-vertex line graph, then rep$(G)\ge{1\over4}n^{1/3}$. Among line graphs of trees, the minimum repetition number is $\Theta(n^{1/2})$. For line graphs of maximal outerplanar graphs, trees with perfect matchings, or triangulations with 2-factors, the lower bound is linear.

scholarly journals Some New Classes of Open Distance-Pattern Uniform Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Bibin K. Jose
Keyword(s):  
Nonempty Subset ◽  
Chordal Graphs ◽  
Outerplanar Graph ◽  
Outerplanar Graphs ◽  
Induced Subgraphs ◽  
Minimum Cardinality ◽  
Maximal Outerplanar Graphs ◽  
Split Graphs ◽  
Strongly Chordal Graphs

Given an arbitrary nonempty subset M of vertices in a graph G=(V,E), each vertex u in G is associated with the set fMo(u)={d(u,v):v∈M,u≠v} and called its open M-distance-pattern. The graph G is called open distance-pattern uniform (odpu-) graph if there exists a subset M of V(G) such that fMo(u)=fMo(v) for all u,v∈V(G), and M is called an open distance-pattern uniform (odpu-) set of G. The minimum cardinality of an odpu-set in G, if it exists, is called the odpu-number of G and is denoted by od(G). Given some property P, we establish characterization of odpu-graph with property P. In this paper, we characterize odpu-chordal graphs, and thereby characterize interval graphs, split graphs, strongly chordal graphs, maximal outerplanar graphs, and ptolemaic graphs that are odpu-graphs. We also characterize odpu-self-complementary graphs, odpu-distance-hereditary graphs, and odpu-cographs. We prove that the odpu-number of cographs is even and establish that any graph G can be embedded into a self-complementary odpu-graph H, such that G and G¯ are induced subgraphs of H. We also prove that the odpu-number of a maximal outerplanar graph is either 2 or 5.

scholarly journals Line Graphs of Monogenic Semigroup Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Nihat Akgunes ◽  
Yasar Nacaroglu ◽  
Sedat Pak
Keyword(s):  
Maximum Degree ◽  
Zero Divisor ◽  
Minimum Degree ◽  
Line Graph ◽  
Domination Number ◽  
Clique Number ◽  
Line Graphs ◽  
Monogenic Semigroup ◽  
Graph Properties ◽  
Graph Parameters

The concept of monogenic semigroup graphs Γ S M is firstly introduced by Das et al. (2013) based on zero divisor graphs. In this study, we mainly discuss the some graph properties over the line graph L Γ S M of Γ S M . In detail, we prove the existence of graph parameters, namely, radius, diameter, girth, maximum degree, minimum degree, chromatic number, clique number, and domination number over L Γ S M .

scholarly journals A Note on Repeated Degrees of Line Graphs

10.37236/9608 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Shimon Kogan
Keyword(s):  
Line Graph ◽  
Vertex Degree ◽  
Line Graphs ◽  
Maximum Multiplicity

Let $\text{rep}(G)$ be the maximum multiplicity of a vertex degree in graph $G$. It was proven in Caro and West [E-JC, 2009] that if $G$ is an $n$-vertex line graph, then $\text{rep}(G) \geqslant \frac{1}{4} n^{1/3}$. In this note we prove that for infinitely many $n$ there is a $n$-vertex line graph $G$ such that $\text{rep}(G) \leqslant \left(2n\right)^{1/3}$, thus showing that the bound above is asymptotically tight. Previously it was only known that for infinitely many $n$ there is a $n$-vertex line graph $G$ such that $\text{rep}(G) \leqslant \sqrt{4n/3}$ (Caro and West [E-JC, 2009]). Finally we prove that if $G$ is a $n$-vertex line graph, then $\text{rep}(G) \geqslant \left(\left(\frac{1}{2}-o(1)\right)n\right)^{1/3}$.

scholarly journals Generalized Line Graphs: Cartesian Products and Complexity of Recognition

10.37236/3983 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Aparna Lakshmanan S. ◽  
Csilla Bujtás ◽  
Zsolt Tuza
Keyword(s):  
Line Graph ◽  
Cartesian Product ◽  
General Setting ◽  
Usual Sense ◽  
Line Graphs ◽  
Free Graph ◽  
Cartesian Products ◽  
Triangle Graph ◽  
Np Complete ◽  
Necessary And Sufficient

Putting the concept of line graph in a more general setting, for a positive integer $k$, the $k$-line graph $L_k(G)$ of a graph $G$ has the $K_k$-subgraphs of $G$ as its vertices, and two vertices of $L_k(G)$ are adjacent if the corresponding copies of $K_k$ in $G$ share $k-1$ vertices. Then, 2-line graph is just the line graph in usual sense, whilst 3-line graph is also known as triangle graph. The $k$-anti-Gallai graph $\triangle_k(G)$ of $G$ is a specified subgraph of $L_k(G)$ in which two vertices are adjacent if the corresponding two $K_k$-subgraphs are contained in a common $K_{k+1}$-subgraph in $G$.We give a unified characterization for nontrivial connected graphs $G$ and $F$ such that the Cartesian product $G\Box F$ is a $k$-line graph. In particular for $k=3$, this answers the question of Bagga (2004), yielding the necessary and sufficient condition that $G$ is the line graph of a triangle-free graph and $F$ is a complete graph (or vice versa). We show that for any $k\ge 3$, the $k$-line graph of a connected graph $G$ is isomorphic to the line graph of $G$ if and only if $G=K_{k+2}$. Furthermore, we prove that the recognition problem of $k$-line graphs and that of $k$-anti-Gallai graphs are NP-complete for each $k\ge 3$.

scholarly journals Topological invariants for the line graphs of some classes of graphs

2019 ◽  
Vol 17 (1) ◽  
pp. 1483-1490
Author(s):  
Xiaoqing Zhou ◽  
Mustafa Habib ◽  
Tariq Javeed Zia ◽  
Asim Naseem ◽  
Anila Hanif ◽  
...  
Keyword(s):  
Communication Theory ◽  
Chemical Reactivity ◽  
Line Graph ◽  
Chemical Properties ◽  
Topological Invariants ◽  
Line Graphs ◽  
Physical And Chemical Properties ◽  
Physical And Chemical ◽  
Ladder Graphs ◽  
And Chemical Properties

AbstractGraph theory plays important roles in the fields of electronic and electrical engineering. For example, it is critical in signal processing, networking, communication theory, and many other important topics. A topological index (TI) is a real number attached to graph networks and correlates the chemical networks with physical and chemical properties, as well as with chemical reactivity. In this paper, our aim is to compute degree-dependent TIs for the line graph of the Wheel and Ladder graphs. To perform these computations, we first computed M-polynomials and then from the M-polynomials we recovered nine degree-dependent TIs for the line graph of the Wheel and Ladder graphs.

scholarly journals An Ordered Turán Problem for Bipartite Graphs

10.37236/2471 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Craig Timmons
Keyword(s):  
Complete Bipartite Graph ◽  
Bipartite Graphs ◽  
Free Graph ◽  
The Family ◽  
Turan Problem ◽  
Vertex Graph ◽  
Turán Number ◽  
Turán Problem ◽  
Natural Way

Let $F$ be a graph.  A graph $G$ is $F$-free if it does not contain $F$ as a subgraph.  The Turán number of $F$, written $\textrm{ex}(n,F)$, is the maximum number of edges in an $F$-free graph with $n$ vertices.  The determination of Turán numbers of bipartite graphs is a challenging and widely investigated problem.  In this paper we introduce an ordered version of the Turán problem for bipartite graphs.  Let $G$ be a graph with $V(G) = \{1, 2, \dots , n \}$ and view the vertices of $G$ as being ordered in the natural way.  A zig-zag $K_{s,t}$, denoted $Z_{s,t}$, is a complete bipartite graph $K_{s,t}$ whose parts $A = \{n_1 < n_2 < \dots < n_s \}$ and $B = \{m_1 < m_2 < \dots < m_t \}$ satisfy the condition $n_s < m_1$.  A zig-zag $C_{2k}$ is an even cycle $C_{2k}$ whose vertices in one part precede all of those in the other part.  Write $\mathcal{Z}_{2k}$ for the family of zig-zag $2k$-cycles.  We investigate the Turán numbers $\textrm{ex}(n,Z_{s,t})$ and $\textrm{ex}(n,\mathcal{Z}_{2k})$.  In particular we show $\textrm{ex}(n, Z_{2,2}) \leq \frac{2}{3}n^{3/2} + O(n^{5/4})$.  For infinitely many $n$ we construct a $Z_{2,2}$-free $n$-vertex graph with more than $(n - \sqrt{n} - 1) + \textrm{ex} (n,K_{2,2})$ edges.

scholarly journals Another Infinite Sequence of Dense Triangle-Free Graphs

10.37236/1381 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Stephan Brandt ◽  
Tomaž Pisanski
Keyword(s):  
Structural Properties ◽  
Chromatic Number ◽  
Infinite Sequence ◽  
Minimum Degree ◽  
Free Graph ◽  
The Core ◽  
Graph Homomorphisms ◽  
Natural Way

The core is the unique homorphically minimal subgraph of a graph. A triangle-free graph with minimum degree $\delta > n/3$ is called dense. It was observed by many authors that dense triangle-free graphs share strong structural properties and that the natural way to describe the structure of these graphs is in terms of graph homomorphisms. One infinite sequence of cores of dense maximal triangle-free graphs was known. All graphs in this sequence are 3-colourable. Only two additional cores with chromatic number 4 were known. We show that the additional graphs are the initial terms of a second infinite sequence of cores.

scholarly journals Lower Bounds for the Size of Random Maximal $H$-Free Graphs

10.37236/93 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Guy Wolfovitz
Keyword(s):  
Lower Bound ◽  
Random Process ◽  
Lower Bounds ◽  
Random Permutation ◽  
Bipartite Graphs ◽  
Expected Number ◽  
Free Graph ◽  
Vertex Graph

We consider the next random process for generating a maximal $H$-free graph: Given a fixed graph $H$ and an integer $n$, start by taking a uniformly random permutation of the edges of the complete $n$-vertex graph $K_n$. Then, traverse the edges of $K_n$ according to the order imposed by the permutation and add each traversed edge to an (initially empty) evolving $n$-vertex graph - unless its addition creates a copy of $H$. The result of this process is a maximal $H$-free graph ${\Bbb M}_n(H)$. Our main result is a new lower bound on the expected number of edges in ${\Bbb M}_n(H)$, for $H$ that is regular, strictly $2$-balanced. As a corollary, we obtain new lower bounds for Turán numbers of complete, balanced bipartite graphs. Namely, for fixed $r \ge 5$, we show that ex$(n, K_{r,r}) = \Omega(n^{2-2/(r+1)}(\ln\ln n)^{1/(r^2-1)})$. This improves an old lower bound of Erdős and Spencer. Our result relies on giving a non-trivial lower bound on the probability that a given edge is included in ${\Bbb M}_n(H)$, conditioned on the event that the edge is traversed relatively (but not trivially) early during the process.

scholarly journals A Strengthening of Brooks' Theorem for Line Graphs

10.37236/632 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Landon Rabern
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Line Graph ◽  
Clique Number ◽  
Line Graphs ◽  
Delta G

We prove that if $G$ is the line graph of a multigraph, then the chromatic number $\chi(G)$ of $G$ is at most $\max\left\{\omega(G), \frac{7\Delta(G) + 10}{8}\right\}$ where $\omega(G)$ and $\Delta(G)$ are the clique number and the maximum degree of $G$, respectively. Thus Brooks' Theorem holds for line graphs of multigraphs in much stronger form. Using similar methods we then prove that if $G$ is the line graph of a multigraph with $\chi(G) \geq \Delta(G) \geq 9$, then $G$ contains a clique on $\Delta(G)$ vertices. Thus the Borodin-Kostochka Conjecture holds for line graphs of multigraphs.

scholarly journals On edge-intersection graphs of k-bend paths in grids

2010 ◽  
Vol Vol. 12 no. 1 ◽  
Author(s):  
Therese Biedl ◽  
Michal Stern
Keyword(s):  
Conflict Resolution ◽  
Planar Graph ◽  
Bipartite Graph ◽  
Outerplanar Graph ◽  
Line Graphs ◽  
Intersection Graphs ◽  
Grid Networks ◽  
Graph Classes ◽  
International Audience ◽  
Number Of Bends

International audience Edge-intersection graphs of paths in grids are graphs that can be represented such that vertices are paths in a grid and edges between vertices of the graph exist whenever two grid paths share a grid edge. This type of graphs is motivated by applications in conflict resolution of paths in grid networks. In this paper, we continue the study of edge-intersection graphs of paths in a grid, which was initiated by Golumbic, Lipshteyn and Stern. We show that for any k, if the number of bends in each path is restricted to be at most k, then not all graphs can be represented. Then we study some graph classes that can be represented with k-bend paths, for small k. We show that every planar graph has a representation with 5-bend paths, every outerplanar graph has a representation with 3-bend paths, and every planar bipartite graph has a representation with 2-bend paths. We also study line graphs, graphs of bounded pathwidth, and graphs with -regular edge orientations.

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