Let ${\mu }_{n-1}\left(G\right)$ be the algebraic connectivity, and let ${\mu }_1\left(G\right)$ be the Laplacian spectral radius of a $k$-connected graph $G$ with $n$ vertices and $m$ edges. In this paper, we prove that $${\mu }_{n-1}\left(G\right)\ge \frac{2n{k}^2}{(n(n-1)-2m)(n+k-2)+2k^2},$$ with equality if and only if $G$ is the complete graph $K_n$ or $K_n-e$. Moreover, if $G$ is non-regular, then $${\mu }_1\left(G\right)<2\Delta -\frac{2(n\Delta -2m)k^2}{2(n\Delta -2m)(n^2-2n+2k)+n{k}^2},$$ where $\Delta$ stands for the maximum degree of $G$. Remark that in some cases, these two inequalities improve some previously known results.

The imbalance of an edge $e=\{u,v\}$ in a graph is defined as $i\left(e\right)=\left|d\right(u)-d(v\left)\right|$, where $d(·)$ is the vertex degree. The irregularity $I\left(G\right)$ of $G$ is then defined as the sum of imbalances over all edges of $G$. This concept was introduced by Albertson who proved that $I\left(G\right)\le 4n^3/27$ (where $n=\left|V\right(G\left)\right|$) and obtained stronger bounds for bipartite and triangle-free graphs. Since then a number of additional bounds were given by various authors. In this paper we prove a new upper bound, which improves a bound found by Zhou and Luo in 2008. Our bound involves...

A signed graph $\Gamma$ is a graph whose edges are labeled by signs. If $\Gamma$ has $n$ vertices, its spectral radius is the number $\rho \left(\Gamma \right):=max\left\{\right|{\lambda }_i\left(\Gamma \right)|:1\le i\le n\}$, where ${\lambda }_1\left(\Gamma \right)\ge \cdots \ge {\lambda }_n\left(\Gamma \right)$ are the eigenvalues of the signed adjacency matrix $A\left(\Gamma \right)$. Here we determine the signed graphs achieving the minimal or the maximal spectral radius in the classes ${𝔘}_n$ and ${𝔅}_n$ of unbalanced unicyclic graphs and unbalanced bicyclic graphs, respectively.

Let $G$ be a graph of order $n$ and $\lambda \left(G\right)$ the spectral radius of its adjacency matrix. We extend some recent results on sufficient conditions for Hamiltonian paths and cycles in $G$. One of the main results of the paper is the following theorem: Let $k\ge 2,$ $n\ge k^3+k+4,$ and let $G$ be a graph of order $n$, with minimum degree $\delta \left(G\right)\ge k.$ If $$\lambda \left(G\right)\ge n-k-1,$$ then $G$ has a Hamiltonian cycle, unless $G=K_1\vee (K_{n-k-1}+K_k)$ or $G=K_k\vee (K_{n-2k}+{\overline{K}}_k).$

Let $G$ be a connected graph of order $n$ and $U$ a unicyclic graph with the same order. We firstly give a sharp bound for $m_G\left(\mu \right)$, the multiplicity of a Laplacian eigenvalue $\mu$ of $G$. As a straightforward result, $m_U\left(1\right)\le n-2$. We then provide two graph operations (i.e., grafting and shifting) on graph $G$ for which the value of $m_G\left(1\right)$ is nondecreasing. As applications, we get the distribution of $m_U\left(1\right)$ for unicyclic graphs on $n$ vertices. Moreover, for the two largest possible values of $m_U\left(1\right)\in \{n-5,n-3\}$, the corresponding graphs $U$ are...

The line graph of a graph $G$, denoted by $L\left(G\right)$, has $E\left(G\right)$ as its vertex set, where two vertices in $L\left(G\right)$ are adjacent if and only if the corresponding edges in $G$ have a vertex in common. For a graph $H$, define ${\overline{\sigma }}_2\left(H\right)=min\{d\left(u\right)+d\left(v\right):uv\in E\left(H\right)\}$. Let $H$ be a 2-connected claw-free simple graph of order $n$ with $\delta \left(H\right)\ge 3$. We show that, if ${\overline{\sigma }}_2\left(H\right)\ge \frac{1}{7}(2n-5)$ and $n$ is sufficiently large, then either $H$ is traceable or the Ryjáček’s closure $\mathrm{cl}\left(H\right)=L\left(G\right)$, where $G$ is an essentially $2$-edge-connected triangle-free graph that can be contracted to one of the two graphs of order 10...

A graph $G$ is $H$-saturated if it contains no $H$ as a subgraph, but does contain $H$ after the addition of any edge in the complement of $G$. The saturation number, $\mathrm{sat}(n,H)$, is the minimum number of edges of a graph in the set of all $H$-saturated graphs of order $n$. We determine the saturation number $\mathrm{sat}(n,P_6+t{P}_2)$ for $n\ge \frac{10}{3}t+10$ and characterize the extremal graphs for $n>\frac{10}{3}t+20$.

An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. Let $G$ be a bridgeless simple graph with minimum degree at least $3$. Jackson and Yoshimoto (2007) showed that $G$ has an even factor containing two arbitrary prescribed edges. They also proved that $G$ has an even factor in which each component has order at least four. Moreover, Xiong, Lu and Han (2009) showed that for each pair of edges $e_1$ and $e_2$ of $G$, there is an even factor containing $e_1$ and $e_2$...

We give a graph theoretic interpretation of $r$-Lah numbers, namely, we show that the $r$-Lah number ${⌊\genfrac{}{}{0pt}{}{n}{k}⌋}_r$ counting the number of $r$-partitions of an $(n+r)$-element set into $k+r$ ordered blocks is just equal to the number of matchings consisting of $n-k$ edges in the complete bipartite graph with partite sets of cardinality $n$ and $n+2r-1$ ($0\le k\le n$, $r\ge 1$). We present five independent proofs including a direct, bijective one. Finally, we close our work with a similar result for $r$-Stirling numbers of the second kind. ...