# Spectral radius and Hamiltonicity of graphs with large minimum degree:
revisited

Research paper by **Jun Ge, Bo Ning**

Indexed on: **01 Jul '16**Published on: **01 Jul '16**Published in: **Mathematics - Combinatorics**

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#### Abstract

Investigating the relationship between eigenvalues of a graph and its cycle
structure is a standard topic in spectral graph theory. This paper mainly
concerns spectral conditions for Hamilton cycles in graphs and in balanced
bipartite graphs, respectively. Our main results are written as:
(1) Let $k\geq 1$, $n\geq \max\{\frac{1}{2}k^3+k+4,6k+5\}$, and let $G$ be a
graph of order $n$, with minimum degree $\delta(G)\geq k$. If its spectral
radius $\lambda(G)\geq n-k-1$, then $G$ has a Hamilton cycle unless $G=K_1\vee
(K_k+K_{n-k-1})$, or $G=K_k\vee (kK_1+K_{n-k-1})$.
(2) Let $k\geq 1$, $n\geq k^3+2k+4$, and let $G$ be a balanced bipartite
graph of order $2n$, with minimum degree $\delta(G)\geq k$. If its spectral
radius $\lambda(G)\geq \sqrt{n(n-k)}$, then $G$ has a Hamilton cycle unless
$G=B^k_n$, where $B^k_n$ is the graph obtained from $K_{n,n}$ by deleting a
$K_{k,n-k}$.
Our first theorem shows that a very recent theorem of Nikiforov still holds
when $n<k^3$ (if $k$ is sufficiently large). Our second result further gives a
spectral analogue of Moon-Moser's theorem on Hamilton cycles in balanced
bipartite graphs, and extends a previous result due to Li and one of the
authors here for $n$ sufficiently large.