Hamiltonian s-properties and eigenvalues of k-connected graphs

Chvátal and Erdös (1972) [5] proved that, for a k-connected graph G, if the stability number $\alpha (G)\le k?s$, then G is Hamilton-connected ($s=1$) or Hamiltonian ($s=0$) or traceable ($s=?1$). Motivated by the result, we focus on tight sufficient spectral conditions for k-connected graphs to possess Hamiltonian s-properties. We say that a graph possesses Hamiltonian s-properties, which means that the graph is Hamilton-connected if $s=1$, Hamiltonian if $s=0$, and traceable if $s=?1$.For a real number $a\ge 0$, and for a k-connected graph G with order n, degree diagonal matrix $D(G)$ and adjacency matrix $A(G)$, we have identified best possible upper bounds for the spectral radius ${\lambda }_1(aD(\mathrm{\Gamma })+A(\mathrm{\Gamma }))$, where Γ is either G or the complement of G, to warrant that G possesses Hamiltonian s-properties. Sufficient conditions for a graph G to possess Hamiltonian s-properties in terms of upper bounds for the Laplacian spectral radius as well as lower bounds of the algebraic connectivity of G are also obtained. Other best possible spectral conditions for Hamiltonian s-properties are also discussed.     

Graphs with nullity 2c(G)+p(G)−1

Let G be a connected graph with $n(G)$ vertices and $e(G)$ edges. The nullity of G, denoted by $\eta (G)$, is the multiplicity of eigenvalue zero of the adjacency matrix of G. Ma, Wong and Tian (2016) proved that $\eta (G)\le 2c(G)+p(G)?1$ unless G is a cycle of order a multiple of 4, where $c(G)=e(G)?n(G)+1$ is the elementary cyclic number of G and $p(G)$ is the number of leaves of G. Recently, Chang, Chang and Zheng (2020) characterized the leaf-free graphs with nullity $2c(G)?1$, thus leaving the problem to characterize connected graphs G with nullity $2c(G)+p(G)?1$ when $p(G)\ne 0$. In this paper, we solve this problem completely.     

An improvement to the Hilton-Zhao vertex-splitting conjecture

For a simple graph G, denote by n, $\mathrm{\Delta }(G)$, and ${\chi }^{\prime }(G)$ its order, maximum degree, and chromatic index, respectively. A graph G is edge-chromatic critical if ${\chi }^{\prime }(G)=\mathrm{\Delta }(G)+1$ and ${\chi }^{\prime }(H)<{\chi }^{\prime }(G)$ for every proper subgraph H of G. Let G be an n-vertex connected regular class 1 graph, and let $G^{?}$ be obtained from G by splitting one vertex of G into two vertices. Hilton and Zhao in 1997 conjectured that $G^{?}$ must be edge-chromatic critical if $\mathrm{\Delta }(G)>n/3$, and they verified this when $\mathrm{\Delta }(G)\ge \frac{n}{2}(\sqrt{7}?1)\approx 0.82n$. In this paper, we prove it for $\mathrm{\Delta }(G)\ge 0.75n$.     

On the size of special class 1 graphs and (P3;k)-co-critical graphs

A well-known theorem of Vizing states that if G is a simple graph with maximum degree Δ, then the chromatic index ${\chi }^{\prime }(G)$ of G is Δ or $\mathrm{\Delta }+1$. A graph G is class 1 if ${\chi }^{\prime }(G)=\mathrm{\Delta }$, and class 2 if ${\chi }^{\prime }(G)=\mathrm{\Delta }+1$; G is Δ-critical if it is connected, class 2 and ${\chi }^{\prime }(G-e)<{\chi }^{\prime }(G)$ for every $e\in E(G)$. A long-standing conjecture of Vizing from 1968 states that every Δ-critical graph on n vertices has at least $(n(\mathrm{\Delta }-1)+3)/2$ edges. We initiate the study of determining the minimum number of edges of class 1 graphs G, in addition, ${\chi }^{\prime }(G+e)={\chi }^{\prime }(G)+1$ for every $e\in E(\bar{G})$. Such graphs have intimate relation to $(P_3;k)$-co-critical graphs, where a non-complete graph G is $(P_3;k)$-co-critical if there exists a k-coloring of $E(G)$ such that G does not contain a monochromatic copy of $P_3$ but every k-coloring of $E(G+e)$ contains a monochromatic copy of $P_3$ for every $e\in E(\bar{G})$. We use the bound on the size of the aforementioned class 1 graphs to study the minimum number of edges over all $(P_3;k)$-co-critical graphs. We prove that if G is a $(P_3;k)$-co-critical graph on $n\ge k+2$ vertices, then$e(G)\ge \frac{k}{2}(n-\lceil \frac{k}{2}\rceil -\epsilon )+\left(\begin{array}{c}\lceil k/2\rceil +\epsilon \\ 2\end{array}\right),$ where ε is the remainder of $n-\lceil k/2\rceil$ when divided by 2. This bound is best possible for all $k\ge 1$ and $n\ge \lceil 3k/2\rceil +2$.     

Hamiltonian and long paths in bipartite graphs with connectivity

Let G be a graph, $\nu (G)$ the order of G, $\kappa (G)$ the connectivity of G and k a positive integer such that $k\le (\nu (G)?2)/2$. Then G is said to be k-extendable if it has a matching of size k and every matching of size k extends to a perfect matching of G. A Hamiltonian path of a graph G is a spanning path of G. A bipartite graph G with vertex sets $V_1$ and $V_2$ is defined to be Hamiltonian-laceable if such that $|V_1|=|V_2|$ and for every pair of vertices $p\in V_1$ and $q\in V_2$, there exists a Hamiltonian path in G with endpoints p and q, or $|V_1|=|V_2|+1$ and for every pair of vertices $p,q\in V_1,p\ne q$, there exists a Hamiltonian path in G with endpoints p and q. Let G be a bipartite graph with bipartition $(X,Y)$. Define $bn(G)$ to be a maximum integer such that $0\le bn(G)<min\{|X|,|Y|\}$ and (1) for each non-empty subset S of X, if $|S|\le |X|?bn(G)$, then $|N(S)|\ge |S|+bn(G)$, and if $|X|?bn(G)<|S|\le |X|$, then $N(S)=Y$; and (2) for each non-empty subset S of Y, if $|S|\le |Y|?bn(G)$, then $|N(S)|\ge |S|+bn(G)$, and if $|Y|?bn(G)<|S|\le |Y|$, then $N(S)=X$; and (3) $bn(G)=0$ if there is no non-negative integer satisfying (1) and (2).Let G be a bipartite graph with bipartition $(X,Y)$ such that $|X|=|Y|$ and $bn(G)>0$. In this paper, we show that if $\nu (G)\le 2\kappa (G)+4bn(G)?4$, then G is Hamiltonian-laceable; or if $\nu (G)>6bn(G)?2$, then for every pair of vertices $x\in X$ and $y\in Y$, there is an $(x,y)$-path P in G with $|V(P)|\ge 6bn(G)?2$. We show some of its corollaries in k-extendable, bipartite graphs and a conjecture in k-extendable graphs.     

Maximize the Q-index of graphs with fixed order and size

In this paper, we consider two kinds of spectral extremal questions. The first asks which graph attains the maximum Q-index over all graphs of order n and size $m=n+k$? The second asks which graph attains the maximum Q-index over all $(p,q)$-bipartite graphs with $m=p+q+k$ edges? We solve the first question for $4\le k\le n?3$, and the second question for $?p?q\le k\le p?q$. The maximum Q-index on connected $(p,q)$-bipartite graphs is also determined for $?1\le k\le p?2$.     

A characterization of graphs with given maximum degree and smallest possible matching number: II

Let ${\alpha }^{\prime }(G)$ be the matching number of a graph G. A characterization of the graphs with given maximum odd degree and smallest possible matching number is given by Henning and Shozi (2021) [13]. In this paper we complete our study by giving a characterization of the graphs with given maximum even degree and smallest possible matching number. In 2018 Henning and Yeo [10] proved that if G is a connected graph of order n, size m and maximum degree k where $k\ge 4$ is even, then ${\alpha }^{\prime }(G)\ge \frac{n}{k(k+1)}+\frac{m}{k+1}?\frac{1}{k(k+1)}$, unless G is k-regular and $n\in \{k+1,k+3\}$. In this paper, we give a complete characterization of the graphs that achieve equality in this bound when the maximum degree k is even, thereby completing our study of graphs with given maximum degree and smallest possible matching number.     

Hamilton-connected,vertex-pancyclic and bipartite holes

In 2016, McDiarmid and Yolov gave a tight threshold for the existence of Hamilton cycle in graphs with large minimum degree and without large “bipartite hole”(two disjoint sets of vertices with no edge between them) which extends Dirac's classical Theorem. In detail, an $(s,t)$-bipartite-hole in a graph G consists of two disjoint vertex sets S and T with $|S|=s$ and $|T|=t$ such that $E(G[S,T])=?$. Let $\stackrel{?}{\alpha }(G)$ be the maximum integer such that G contains an $(s,t)$-bipartite-hole for every pair of nonnegative integers s and t with $s+t=r$. Motivated by Bondy's metaconjecture, in this paper, we study the existence of vertex-pancyclicity (every vertex is in a cycle of length i for each $i\in [3,n]$ and Hamilton-connectivity(any two vertices can be connected through a Hamilton path). Our central theorem is that for any given $\mu >0$ and sufficiently large n, if G is an n-vertex graph with $\stackrel{?}{\alpha }(G)=\mu n$ and $\delta (G)\ge \frac{10}{3}\mu n$, then G is Hamilton-connected and vertex-pancyclic.     

A note on the distribution of weights of fixed-rank matrices over the binary field

Let M be a random $m\times n$ rank-r matrix over the binary field ${\mathbb{F}}_2$, and let $\mathrm{wt}(\mathbf{M})$ be its Hamming weight, that is, the number of nonzero entries of M.We prove that, as $m,n\to +\infty$ with r fixed and $m/n$ tending to a constant, we have that$\frac{\mathrm{wt}(\mathbf{M})-\frac{1-2^{-r}}{2}mn}{\sqrt{\frac{2^{-r}(1-2^{-r})}{4}(m+n)mn}}$ converges in distribution to a standard normal random variable.     

On Serre dimension of monoid algebras and Segre extensions

Let R be a commutative noetherian ring of dimension d and M be a commutative, cancellative, torsion-free monoid of rank r. Then S-$dim(R[M])\le max\{1,dim(R[M])?1\}$. Further, we define a class of monoids ${\{{\mathfrak{M}}_n\}}_{n\ge 1}$ such that if $M\in {\mathfrak{M}}_n$ is seminormal, then S-$dim(R[M])\le dim(R[M])?n=d+r?n$, where $1\le n\le r$. As an application, we prove that for the Segre extension $S_{mn}(R)$ over R, S-$dim(S_{mn}(R))\le dim(S_{mn}(R))?\left[\frac{m+n?1}{min\{m,n\}}\right]=d+m+n?1?\left[\frac{m+n?1}{min\{m,n\}}\right]$.