Some Relations between Rank, Chromatic Number and Energy of Graphs

Details Some Relations between Rank, Chromatic Number and Energy of Graphs Some Relations between Rank, Chromatic Number and Energy of Graphs

2007-09-20

arxiv.org/abs/0709.3140v1

Mathematics

Combinatorics

Accepted for publication in Discrete Mathematics

Scientific paper

The energy of a graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of $G$. Let $G$ be a graph of order $n$ and ${\rm rank}(G)$ be the rank of the adjacency matrix of $G$. In this paper we characterize all graphs with $E(G)={\rm rank}(G)$. Among other results we show that apart from a few families of graphs, $E(G)\geq 2\max(\chi(G), n-\chi(\bar{G}))$, where $n$ is the number of vertices of $G$, $\bar{G}$ and $\chi(G)$ are the complement and the chromatic number of $G$, respectively. Moreover some new lower bounds for $E(G)$ in terms of ${\rm rank}(G)$ are given.

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