The relation between Hamiltonicity and toughness of a graph is a long standing research problem. The paper studies the Hamiltonicity of the Cartesian product graph $G_1square G_2$ of graphs $G_1$ and $G_2$ satisfying that $G_1$ is traceable and $G_2$
is connected with a path factor. Let Pn be the path of order $n$ and $H$ be a connected bipartite graph. With certain requirements of $n$, we show that the following three statements are equivalent: (i) $P_nsquare H$ is Hamiltonian; (ii) $P_nsquare H$ is $1$-tough; and (iii) $H$ has a path factor.
We realize many sharp spectral bounds of the spectral radius of a nonnegative square matrix $C$ by using the largest real eigenvalues of suitable matrices of smaller sizes related to $C$ that are very easy to find. As applications, we give a sharp up
per bound of the spectral radius of $C$ expressed by the sum of entries, the largest off-diagonal entry $f$ and the largest diagonal entry $d$ in $C$. We also give a new class of sharp lower bounds of the spectral radius of $C$ expressed by the above $d$ and $f$, the least row-sum $r_n$ and the $t$-th largest row-sum $r_t$ in $C$ satisfying $0<r_n-(n-t-1)f-dleq r_t-(n-t)f$, where $n$ is the size of $C$.
It is not hard to find many complete bipartite graphs which are not determined by their spectra. We show that the graph obtained by deleting an edge from a complete bipartite graph is determined by its spectrum. We provide some graphs, each of which
is obtained from a complete bipartite graph by adding a vertex and an edge incident on the new vertex and an original vertex, which are not determined by their spectra.
Let $G$ denote a bipartite graph with $e$ edges without isolated vertices. It was known that the spectral radius of $G$ is at most the square root of $e$, and the upper bound is attained if and only if $G$ is a complete bipartite graph. Suppose that
$G$ is not a complete bipartite graph, and $e-1$ and $e+1$ are not twin primes. We determine the maximal spectral radius of $G$. As a byproduct of our study, we obtain a spectral characterization of a pair $(e-1, e+1)$ of integers to be a pair of twin primes.
The Laplacian spread of a graph is the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. We find that the class of strongly regular graphs attains the maximum of largest eigenvalues, th
e minimum of second-smallest eigenvalues of Laplacian matrices and hence the maximum of Laplacian spreads among all simple connected graphs of fixed order, minimum degree, maximum degree, minimum size of common neighbors of two adjacent vertices and minimum size of common neighbors of two nonadjacent vertices. Some other extremal graphs are also provided.
Let k, p, q be positive integers with k < p < q+1. We prove that the maximum spectral radius of a simple bipartite graph obtained from the complete bipartite graph Kp,q of bipartition orders p and q by deleting k edges is attained when the deleting e
dges are all incident on a common vertex which is located in the partite set of order q. Our method is based on new sharp upper bounds on the spectral radius of bipartite graphs in terms of their degree sequences.
Let G be a simple connected graph of order n with degree sequence d_1, d_2, ..., d_n in non-increasing order. The spectral radius rho(G) of G is the largest eigenvalue of its adjacency matrix. For each positive integer L at most n, we give a sharp up
per bound for rho(G) by a function of d_1, d_2, ..., d_L, which generalizes a series of previous results.
Let $W$ denote a simply-laced Coxeter group with $n$ generators. We construct an $n$-dimensional representation $phi$ of $W$ over the finite field $F_2$ of two elements. The action of $phi(W)$ on $F_2^n$ by left multiplication is corresponding to a c
ombinatorial structure extracted and generalized from Vogan diagrams. In each case W of types A, D and E, we determine the orbits of $F_2^n$ under the action of $phi(W)$, and find that the kernel of $phi$ is the center $Z(W)$ of $W.$
Let $X=(V,E)$ be a finite simple connected graph with $n$ vertices and $m$ edges. A configuration is an assignment of one of two colors, black or white, to each edge of $X.$ A move applied to a configuration is to select a black edge $epsilonin E$ an
d change the colors of all adjacent edges of $epsilon.$ Given an initial configuration and a final configuration, try to find a sequence of moves that transforms the initial configuration into the final configuration. This is the edge-flipping puzzle on $X,$ and it corresponds to a group action. This group is called the edge-flipping group $mathbf{W}_E(X)$ of $X.$ This paper shows that if $X$ has at least three vertices, $mathbf{W}_E(X)$ is isomorphic to a semidirect product of $(mathbb{Z}/2mathbb{Z})^k$ and the symmetric group $S_n$ of degree $n,$ where $k=(n-1)(m-n+1)$ if $n$ is odd, $k=(n-2)(m-n+1)$ if $n$ is even, and $mathbb{Z}$ is the additive group of integers.
Let $S$ be a connected graph which contains an induced path of $n-1$ vertices, where $n$ is the order of $S.$ We consider a puzzle on $S$. A configuration of the puzzle is simply an $n$-dimensional column vector over ${0, 1}$ with coordinates of the
vector indexed by the vertex set $S$. For each configuration $u$ with a coordinate $u_s=1$, there exists a move that sends $u$ to the new configuration which flips the entries of the coordinates adjacent to $s$ in $u.$ We completely determine if one configuration can move to another in a sequence of finite steps.
