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A graph $G$ is $H$-saturated if $H$ is not a subgraph of $G$ but $H$ is a subgraph of $G + e$ for any edge $e$ in $\overline{G}$. The saturation number $sat(n,H)$ for a graph $H$ is the minimal number of edges in any $H$-saturated graph of order $n$. The $sat(n, K_{p_1} \cup K_{p_2} \cup K_{p_3})$ with $p_3 \ge p_1 + p_2$ was given in [Discrete Math. 347 (2024) 113868]. In this paper, $sat(n,K_{p_1} \cup K_{p_2} \cup K_{p_3} \cup K_{p_4})$ with $p_{i+1} - p_i \ge p_1$ for $2 \le i\le 3$ and $4\le p_1\le p_2$ is determined., Comment: 17pages

Let $H$ be a fixed graph, a graph G is $H$-saturated if it has no copy of $H$ in $G$, but the addition of any edge in $E(\overline G)$ to $G$ results in an $H$-subgraph. The saturation number sat$(n,H)$ is the minimum number of edges in an $H$-saturated graph on $n$ vertices. In this paper, we determine the saturation number sat$(n,P_7+tP_2)$ for $n\geq \frac {14}{5}t+27$ and characterize the extremal graphs for $n\geq \frac{14}{13}(3t+25)$., Comment: 11 pages

Let $H$ be a fixed graph. A graph $G$ is called {\it $H$-saturated} if $H$ is not a subgraph of $G$ but the addition of any missing edge to $G$ results in an $H$-subgraph. The {\it saturation number} of $H$, denoted $sat(n,H)$, is the minimum number of edges over all $H$-saturated graphs of order $n$, and $Sat(n,H)$ denote the family of $H$-saturated graphs with $sat(n,H)$ edges and $n$ vertices. In this paper, we resolve a conjecture of Chen and Yuan in[Discrete Math. 347(2024)113868] by determining $Sat(n,K_p\cup (t-1)K_q)$ for every $2\le p\le q$ and $t\ge 2$., Comment: 7 pages, 2 figures

Let $S\subseteq V(G)$ and $\pi_{G}(S)$ denote the maximum number $t$ of edge-disjoint paths $P_{1},P_{2},\ldots,P_{t}$ in a graph $G$ such that $V(P_{i})\cap V(P_{j})=S$ for any $i,j\in\{1,2,\ldots,t\}$ and $i\neq j$. If $S=V(G)$, then $\pi_{G}(S)$ is the maximum number of edge-disjoint spanning paths in $G$. It is proved [Graphs Combin., 37 (2021) 2521-2533] that deciding whether $\pi_G(S)\geq r$ is NP-complete for a given $S\subseteq V(G)$. For an integer $r$ with $2\leq r\leq n$, the $r$-path connectivity of a graph $G$ is defined as $\pi_{r}(G)=$min$\{\pi_{G}(S)|S\subseteq V(G)$ and $|S|=r\}$, which is a generalization of tree connectivity. In this paper, we study the $3$-path connectivity of the $k$-dimensional data center network with $n$-port switches $D_{k,n}$ which has significate role in the cloud computing, and prove that $\pi_{3}(D_{k,n})=\lfloor\frac{2n+3k}{4}\rfloor$ with $k\geq 1$ and $n\geq 6$.

A mixed graph $\widetilde{G}$ is obtained from a simple undirected graph $G$, the underlying graph of $\widetilde{G}$, by orienting some edges of $G$. Let $c(G)=|E(G)|-|V(G)|+\omega(G)$ be the cyclomatic number of $G$ with $\omega(G)$ the number of connected components of $G$, $m(G)$ be the matching number of $G$, and $\eta(\widetilde{G})$ be the nullity of $\widetilde{G}$. Chen et al. (2018)\cite{LSC} and Tian et al. (2018)\cite{TFL} proved independently that $|V(G)|-2m(G)-2c(G) \leq \eta(\widetilde{G}) \leq |V(G)|-2m(G)+2c(G)$, respectively, and they characterized the mixed graphs with nullity attaining the upper bound and the lower bound. In this paper, we prove that there is no mixed graph with nullity $\eta(\widetilde{G})=|V(G)|-2m(G)+2c(G)-1$. Moreover, for fixed $c(G)$, there are infinitely many connected mixed graphs with nullity $|V(G)|-2m(G)+2c(G)-s$ $( 0 \leq s \leq 3c(G), s\neq1 )$ is proved.

Let $G$ be a connected graph. The revised edge Szeged index of $G$ is defined as $Sz^{\ast}_{e}(G)=\sum\limits_{e=uv\in E(G)}(m_{u}(e|G)+\frac{m_{0}(e|G)}{2})(m_{v}(e|G)+\frac{m_{0}(e|G)}{2})$, where $m_{u}(e|G)$ (resp., $m_{v}(e|G)$) is the number of edges whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$), and $m_{0}(e|G)$ is the number of edges equidistant from both ends of $e$, respectively. In this paper, the graphs with minimum revised edge Szeged index among all the unicyclic graphs with given diameter are characterized., Comment: arXiv admin note: text overlap with arXiv:1805.06578

The balanced hypercube $BH_{n}$, proposed by Wu and Huang, is a variation of the hypercube. The paired 1-disjoint path cover of $BH_{n}$ is the Hamiltonian laceability, which was obtained by Xu et al. in [Appl. Math. Comput. 189 (2007) 1393--1401]. The paired 2-disjoint path cover of $BH_{n}$ was obtained by Cheng et al. in [Appl. Math. and Comput. 242 (2014) 127-142]. In this paper, we obtain the paired 3-disjoint path cover of $BH_{n}$ with $n\geq 3$. This result improves the above known results about the paired $k$-disjoint path covers of $BH_{n}$ for $k=1,2$., Comment: 15 pages, 5 figures. The work was included in the MS thesis of the author Mei-Rong Guo in [Path Cover and Fault-tolerant Cycle Embedding Analysis of Some Networks, MS Thesis at Beijing Jiaotong University, China, June 2016]

A connected graph $G$ is called strongly Menger (edge) connected if for any two distinct vertices $x,y$ of $G$, there are $\min \{{\rm deg}_G(x), {\rm deg}_G(y)\}$ vertex(edge)-disjoint paths between $x$ and $y$. In this paper, we consider strong Menger (edge) connectedness of the augmented $k$-ary $n$-cube $AQ_{n,k}$, which is a variant of $k$-ary $n$-cube $Q_n^k$. By exploring the topological proprieties of $AQ_{n,k}$, we show that $AQ_{n,3}$ for $n\geq 4$ (resp.\ $AQ_{n,k}$ for $n\geq 2$ and $k\geq 4$) is still strongly Menger connected even when there are $4n-9$ (resp.\ $4n-8$) faulty vertices and $AQ_{n,k}$ is still strongly Menger edge connected even when there are $4n-4$ faulty edges for $n\geq 2$ and $k\geq 3$. Moreover, under the restricted condition that each vertex has at least two fault-free edges, we show that $AQ_{n,k}$ is still strongly Menger edge connected even when there are $8n-10$ faulty edges for $n\geq 2$ and $k\geq 3$. These results are all optimal in the sense of the maximum number of tolerated vertex (resp.\ edge) faults., Comment: 18 pages, 4 figures

A complex unit gain graph (or $\mathbb{T}$-gain graph) is a triple $\Phi=(G, \mathbb{T}, \varphi)$ ($(G, \varphi)$ for short) consisting of a graph $G$ as the underlying graph of $(G, \varphi)$, $\mathbb{T}= \{ z \in C:|z|=1 \} $ is a subgroup of the multiplicative group of all nonzero complex numbers $\mathbb{C}^{\times}$ and a gain function $\varphi: \overrightarrow{E} \rightarrow \mathbb{T}$ such that $\varphi(e_{ij})=\varphi(e_{ji})^{-1}=\overline{\varphi(e_{ji})}$. In this paper, we investigate the relation among the rank, the independence number and the cyclomatic number of a complex unit gain graph $(G, \varphi)$ with order $n$, and prove that $2n-2c(G) \leq r(G, \varphi)+2\alpha(G) \leq 2n$. Where $r(G, \varphi)$, $\alpha(G)$ and $c(G)$ are the rank of the Hermitian adjacency matrix $A(G, \varphi)$, the independence number and the cyclomatic number of $G$, respectively. Furthermore, the properties of the complex unit gain graph that reaching the lower bound are characterized., Comment: arXiv admin note: substantial text overlap with arXiv:1907.07837, arXiv:1909.07555

A mixed graph $\widetilde{G}$ is obtained by orienting some edges of $G$, where $G$ is the underlying graph of $\widetilde{G}$. The positive inertia index, denoted by $p^{+}(G)$, and the negative inertia index, denoted by $n^{-}(G)$, of a mixed graph $\widetilde{G}$ are the integers specifying the numbers of positive and negative eigenvalues of the Hermitian adjacent matrix of $\widetilde{G}$, respectively. In this paper, we study the positive and negative inertia index of the mixed unicyclic graph. Moreover, we give the upper and lower bounds of the positive and negative inertia index of the mixed graph, and characterize the mixed graphs which attain the upper and lower bounds respectively.

A complex unit gain graph (or ${\mathbb T}$-gain graph) is a triple $\Phi=(G, {\mathbb T}, \varphi)$ (or $(G, \varphi)$ for short) consisting of a simple graph $G$, as the underlying graph of $(G, \varphi)$, the set of unit complex numbers $\mathbb{T}= \{ z \in C:|z|=1 \}$ and a gain function $\varphi: \overrightarrow{E} \rightarrow \mathbb{T}$ with the property that $\varphi(e_{i,j})=\varphi(e_{j,i})^{-1}$. In this paper, we prove that $2m(G)-2c(G) \leq r(G, \varphi) \leq 2m(G)+c(G)$, where $r(G, \varphi)$, $m(G)$ and $c(G)$ are the rank of the Hermitian adjacency matrix $H(G, \varphi)$, the matching number and the cyclomatic number of $G$, respectively. Furthermore, the complex unit gain graphs $(G, \mathbb{T}, \varphi)$ with $r(G, \varphi)=2m(G)-2c(G)$ and $r(G, \varphi)=2m(G)+c(G)$ are characterized. These results generalize the corresponding known results about undirected graphs, mixed graphs and signed graphs. Moreover, we show that $2m(G-V_{0}) \leq r(G, \varphi) \leq 2m(G)+b(G)$ holds for any subset $V_0$ of $V(G)$ such that $G-V_0$ is acyclic and $b(G)$ is the minimum integer $|S|$ such that $G-S$ is bipartite for $S \subset V(G)$.

A signed graph $(G, \sigma)$ is a graph with a sign attached to each of its edges, where $G$ is the underlying graph of $(G, \sigma)$. Let $c(G)$, $\alpha(G)$ and $r(G, \sigma)$ be the cyclomatic number, the independence number and the rank of the adjacency matrix of $(G, \sigma)$, respectively. In this paper, we study the relation among the independence number, the rank and the cyclomatic number of a signed graph $(G, \sigma)$ with order $n$, and prove that $2n-2c(G) \leq r(G, \sigma)+2\alpha(G) \leq 2n$. Furthermore, the signed graphs that reaching the lower bound are investigated.

For an integer $\ell\geqslant 2$, the $\ell$-component connectivity of a graph $G$, denoted by $\kappa_{\ell}(G)$, is the minimum number of vertices whose removal from $G$ results in a disconnected graph with at least $\ell$ components or a graph with fewer than $\ell$ vertices. This is a natural generalization of the classical connectivity of graphs defined in term of the minimum vertex-cut and is a good measure of robustness for the graph corresponding to a network. So far, the exact values of $\ell$-connectivity are known only for a few classes of networks and small $\ell$'s. It has been pointed out in~[Component connectivity of the hypercubes, Int. J. Comput. Math. 89 (2012) 137--145] that determining $\ell$-connectivity is still unsolved for most interconnection networks, such as alternating group graphs and star graphs. In this paper, by exploring the combinatorial properties and fault-tolerance of the alternating group graphs $AG_n$ and a variation of the star graphs called split-stars $S_n^2$, we study their $\ell$-component connectivities. We obtain the following results: (i) $\kappa_3(AG_n)=4n-10$ and $\kappa_4(AG_n)=6n-16$ for $n\geqslant 4$, and $\kappa_5(AG_n)=8n-24$ for $n\geqslant 5$; (ii) $\kappa_3(S_n^2)=4n-8$, $\kappa_4(S_n^2)=6n-14$, and $\kappa_5(S_n^2)=8n-20$ for $n\geqslant 4$.