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Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with $n$ vertices is $(1+o(1)) frac{3n^2}{2pi^2}$. This conjecture can be rephrased in terms of the spectral gap as follows: the spectral gap (algebraic connectivity) of a connected $k$-regular graph on $n$ vertices is at least $(1+o(1))frac{2kpi^2}{3n^2}$, and the bound is attained for at least one value of $k$. Based upon previous work of Brand, Guiduli, and Imrich, we prove this conjecture for cubic graphs. We also investigate the structure of quartic (i.e. 4-regular) graphs with the minimum spectral gap among all connected quartic graphs. We show that they must have a path-like structure built from specific blocks.
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We find precise asymptotic estimates for the number of planar maps and graphs with a condition on the minimum degree, and properties of random graphs from these classes. In particular we show that the size of the largest tree attached to the core of a random planar graph is of order c log(n) for an explicit constant c. These results provide new information on the structure of random planar graphs.
This paper presents sufficient conditions for Hamiltonian paths and cycles in graphs. Letting $lambdaleft( Gright) $ denote the spectral radius of the adjacency matrix of a graph $G,$ the main results of the paper are: (1) Let $kgeq1,$ $ngeq k^{3}/2+k+4,$ and let $G$ be a graph of order $n$, with minimum degree $deltaleft( Gright) geq k.$ If [ lambdaleft( Gright) geq n-k-1, ] then $G$ has a Hamiltonian cycle, unless $G=K_{1}vee(K_{n-k-1}+K_{k})$ or $G=K_{k}vee(K_{n-2k}+overline{K}_{k})$. (2) Let $kgeq1,$ $ngeq k^{3}/2+k^{2}/2+k+5,$ and let $G$ be a graph of order $n$, with minimum degree $deltaleft( Gright) geq k.$ If [ lambdaleft( Gright) geq n-k-2, ] then $G$ has a Hamiltonian path, unless $G=K_{k}vee(K_{n-2k-1}+overline {K}_{k+1})$ or $G=K_{n-k-1}+K_{k+1}$ In addition, it is shown that in the above statements, the bounds on $n$ are tight within an additive term not exceeding $2$.
In this article we have derived the minimum order of an odd regular graph such that the graph has no matching. We have observed that how it is different from the case of even regular graphs. We have checked the consistency of the derived result with Petersens theorem.
We prove that the spectral gap of a finite planar graph $X$ is bounded by $lambda_1(X)le C(frac{log(diam X)}{diam X})^2$ where $C$ depends only on the degree of $X$. We then give a sequence of such graphs showing the the above estimate cannot be improved. This yields a negative answer to a question of Benjamini and Curien on the mixing times of the simple random walk on planar graphs.
In this paper, we present a spectral sufficient condition for a graph to be Hamilton-connected in terms of signless Laplacian spectral radius with large minimum degree.
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