Number Of Line Graphs Research Articles

Location domination number of line graph

A set S which dominate and uniquely locate all vertices in the set V(G) – S is a locating dominating set. In this paper location domination number of line graph of Pn, Cn, Sn, Bm, n, F1, n and Wn have been found

Fuzzy Chroamtic Number of Line Graph using α-Cuts

Fuzzy Chroamtic Number of Line Graph using α-Cuts

Convex Quadratic Programming Approach

The problem of determining a maximum matching or whether there exists a perfect matching, is very common in a large variety of applications and as been extensively studied in graph theory. In this paper we start to introduce a characterisation of a family of graphs for which its stability number is determined by convex quadratic programming. The main results connected with the recognition of this family of graphs are also introduced. It follows a necessary and sufficient condition which characterise a graph with a perfect matching and an algorithmic strategy, based on the determination of the stability number of line graphs, by convex quadratic programming, applied to the determination of a perfect matching. A numerical example for the recognition of graphs with a perfect matching is described. Finally, the above algorithmic strategy is extended to the determination of a maximum matching of an arbitrary graph and some related results are presented.

Convex Quadratic Programming Approach to the Maximum Matching Problem

The problem of determining a maximum matching or whether there exists a perfect matching, is very common in a large variety of applications and as been extensively studied in graph theory. In this paper we start to introduce a characterisation of a family of graphs for which its stability number is determined by convex quadratic programming. The main results connected with the recognition of this family of graphs are also introduced. It follows a necessary and sufficient condition which characterise a graph with a perfect matching and an algorithmic strategy, based on the determination of the stability number of line graphs, by convex quadratic programming, applied to the determination of a perfect matching. A numerical example for the recognition of graphs with a perfect matching is described. Finally, the above algorithmic strategy is extended to the determination of a maximum matching of an arbitrary graph and some related results are presented.

The binding number of line graphs and total graphs

D.R. Woodall [7] introduced the concept of the binding number of a graph G , bind ( G ), and proved that bind( G ) (| V(G) | 1)/(| V(G) | ( G )). In this paper, some properties of a graph with bind...

Track number of line graphs

The track number $\tau(G)$ of a graph $G$ is the minimum number of interval graphs whose union is $G$. We show that the track number of the line graph $L(G)$ of a triangle-free graph $G$ is at least $\lg \lg \chi(G) + 1$, where $\chi(G)$ is the chromatic number of $G$. Using this lower bound and two classical Ramsey-theoretic results from literature, we answer two questions posed by Milans, Stolee, and West [J. Combinatorics, 2015] (MSW15). First we show that the track number $\tau(L(K_n))$ of the line graph of the complete graphs $K_n$ is at least $\lg\lg n - o(1)$. This is asymptotically tight and it improves the bound of $\Omega(\lg\lg n/ \lg\lg\lg n)$ in MSW15. Next we show that for a family of graphs $\mathcal{G}$, $\{\tau(L(G)):G \in \mathcal{G}\}$ is bounded if and only if $\{\chi(G):G \in \mathcal{G}\}$ is bounded. This affirms a conjecture in MSW15. All our lower bounds apply even if one enlarges the covering family from the family of interval graphs to the family of chordal graphs.