arXiv subjects

Rong-Xia Hao

Publications and source records attributed to Rong-Xia Hao.

At least 19 recordsLinked to original sources

arXiv2026

Partial-Twuality Polynomials of Paired Matrices

Gross, Mansour, and Tucker~[European Journal of Combinatorics, 95 (2021): 103329] introduced the \emph{partial-twuality polynomials} of ribbon graphs. Recently, Deng, Jin, and Yan generalized the partial-twuality polynomials to the framework of matrix algebra and investigated several of their basic properties. They asked whether there exist matrix operations, called partial duality $δ$ and partial Petrie duality $τ$, on pairs $(M,A)$, where $M$ is a square matrix whose rows and columns are indexed by a finite set $V$ and $A\subseteq V$, such that $δ^2=τ^2=(δτ)^3=id$ and the exponent of the partial-twuality polynomials coincides with some parameter of the matrix obtained by applying $\bullet$ to $(M, A)$. In this paper, we introduce a paired-matrix framework for partial-twuality polynomials over the binary field $\mathbb{GF}(2)$. We prove that there exist two local operations $δ$ and $τ$ on $\bigl((M,I_{|V|}),A\bigr)$ satisfying $δ^2=τ^2=(δτ)^3=id$ and $P_{\langle \bullet \rangle}((M,I_{|V|}),z)=P_{\langle \bullet \rangle}(M,z)$ for $\bullet \in \{δ, τ, δτ, τδ, δτδ\}$, thereby answering their question affirmatively. Finally, we establish a recurrence relation for the partial $\langleδτδ\rangle$-polynomial with respect to an edge. This recurrence enables the computation of the partial $\langleδτδ\rangle$-polynomial for certain bouquets, simple graphs, and simple signed graphs.

math.CO↗

arXiv2026

Partial Petrial Polynomials of Bouquets

Gross, Mansour, and Tucker [European J. Combin., 95 (2021): 103329] introduced the \emph{partial Petrial polynomial} of a ribbon graph $G$, denoted by $^{\partial}{\varepsilon^{\times}_G}(z)$. For a prime bouquet $B_n$, Yan and Li [Discrete Appl. Math., 375 (2025): 281-289] determined $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ when the intersection graph $I(B_n)$ is either the complete graph or a path, and provided an equivalent condition under which the lowest degree of the nonzero coefficient in $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ is $1$. In this paper, we determine $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ when the intersection graph is a cycle. Moreover, we present a complete characterization of the prime bouquets whose lowest nonzero term in $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ is of degree $2$ and determine the partial Petrial polynomial for the prime bouquets. As corollaries, we determine $^{\partial}{\varepsilon^{\times}_{B_n}}(z)$ when $I(B_n)$ is the complete bipartite and tripartite graph.

math.CO↗

arXiv2026

Partial Petrial Polynomials of Ribbon Graphs

Gross, Mansour, and Tucker [European J. Combin., 95 (2021): 103329] introduced the \emph{partial Petrial polynomial} of a ribbon graph $G$, denoted by $^{\partial}{\varepsilon^{\times}_{G}}(z)$. Beck and Mellor proved, in both orientable and non-orientable cases respectively, that the Euler genus of a bouquet equals the rank of a certain matrix over $\mathbb{GF}(2)$. In this paper, we first generalize Beck and Mellor's results from bouquets to all ribbon graphs. Secondly, we give an equivalent representation of the partial Petrial polynomial for all ribbon graphs. Specifically, the partial Petrial polynomial of a ribbon graph $G$ with $n$ vertices is equal to the sum of this polynomial for $2^{n-1}$ distinct bouquets. Moreover, we give the definition of a modified partial Petrial polynomial by assigning coefficients $+1$ or $-1$ to the terms in the partial Petrial polynomial such that the resulting polynomial satisfies the four-term relation for graphs. Finally, we generalize the modified partial Petrial polynomial from bouquets to all signed simple graphs and prove that this polynomial is $4$-invariant, which provides an answer to the problem posed by Lando [J.~Combin.~Theory Ser.~B,~80~(1) (2000): 104-121]: Which of the known graph invariants are $4$-invariants?

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arXiv2025

A perfect matching reciprocity method for embedding multiple hypercubes in an augmented cube: Applications to Hamiltonian decomposition and fault-tolerant Hamiltonicity

This paper focuses on the embeddability of hypercubes in an important class of Cayley graphs, known as augmented cubes. An $n$-dimensional augmented cube $AQ_n$ is constructed by augmenting the $n$-dimensional hypercube $Q_n$ with additional edges, thus making $Q_n$ a spanning subgraph of $AQ_n$. Dong and Wang (2019) first posed the problem of determining the number of $Q_n$-isomorphic subgraphs in $AQ_n$, which still remains open. By exploiting the Cayley properties of $AQ_n$, we establish a lower bound for this number. What's more, we develop a method for constructing pairs of $Q_n$-isomorphic subgraphs in $AQ_n$ with the minimum number of common edges. This is accomplished through the use of reciprocal perfect matchings, a technique that also relies on the Cayley property of $AQ_n$. As an application, we prove that $AQ_n$ admits $n-1$ edge-disjoint Hamiltonian cycles when $n\geq3$ is odd and $n-2$ cycles when $n$ is even, thereby confirming a conjecture by Hung (2015) for the odd case. Additionally, we prove that $AQ_n$ has a fault-free cycle of every even length from $4$ to $2^n$ with up to $4n-8$ faulty edges, when each vertex is incident to at least two fault-free edges. This result not only provides an alternative proof for the fault-tolerant Hamiltonicity of established by Hsieh and Cian (2010), but also extends their work by demonstrating the fault-tolerant bipancyclicity of $AQ_n$.

math.CO↗

arXiv2024

The saturation number for unions of four cliques

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.

math.CO↗

arXiv2024

Saturation Numbers for Linear Forests $P_7+tP_2$

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 $n\geq \frac{14}{13}(3t+25)$.

math.CO↗

arXiv2024

Minimum saturated graphs for unions of cliques

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 $t\ge 2$.

math.CO↗

arXiv2024

Packing internally disjoint Steiner paths of data center networks

Let $S\subseteq V(G)$ and $π_{G}(S)$ 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 $π_{G}(S)$ is the maximum number of edge-disjoint spanning paths in $G$. It is proved [Graphs Combin., 37 (2021) 2521-2533] that deciding whether $π_G(S)\geq r$ 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 $π_{r}(G)=$min$\{π_{G}(S)|S\subseteq V(G)$ 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 $π_{3}(D_{k,n})=\lfloor\frac{2n+3k}{4}\rfloor$ with $k\geq 1$ and $n\geq 6$.

math.CO↗

arXiv2023

No mixed graph with the nullity $η(\widetilde{G})=|V(G)|-2m(G)+2c(G)-1$

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 $c(G)=|E(G)|-|V(G)|+ω(G)$ be the cyclomatic number of $G$ with $ω(G)$ the number of connected components of $G$, $m(G)$ be the matching number of $G$, and $η(\widetilde{G})$ be the nullity of $\widetilde{G}$. Chen et al. (2018)\cite{LSC} and Tian et al. (2018)\cite{TFL} proved independently that $|V(G)|-2m(G)-2c(G) \leq η(\widetilde{G}) \leq |V(G)|-2m(G)+2c(G)$, 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 $η(\widetilde{G})=|V(G)|-2m(G)+2c(G)-1$. 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.

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arXiv2023

The extremal unicyclic graphs of the revised edge Szeged index with given diameter

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.

math.CO↗

arXiv2020

The rank of a complex unit gain graph in terms of the matching number

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

math.CO↗

arXiv2019

Paired 3-disjoint path covers of balanced hypercubes

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 $k=1,2$.

math.CO↗

arXiv2019

Strong Menger connectedness of augmented $k$-ary $n$-cubes

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.

math.CO↗

arXiv2019

Bounds for the rank of a complex unit gain graph in terms of the independence number

A complex unit gain graph (or $\mathbb{T}$-gain graph) is a triple $Φ=(G, \mathbb{T}, φ)$ ($(G, φ)$ for short) consisting of a graph $G$ as the underlying graph of $(G, φ)$, $\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 $φ: \overrightarrow{E} \rightarrow \mathbb{T}$ such that $φ(e_{ij})=φ(e_{ji})^{-1}=\overline{φ(e_{ji})}$. In this paper, we investigate the relation among the rank, the independence number and the cyclomatic number of a complex unit gain graph $(G, φ)$ with order $n$, and prove that $2n-2c(G) \leq r(G, φ)+2α(G) \leq 2n$. Where $r(G, φ)$, $α(G)$ and $c(G)$ are the rank of the Hermitian adjacency matrix $A(G, φ)$, 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.

math.CO↗

arXiv2019

On the inertia index of a mixed graph with the matching number

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.

math.CO↗

arXiv2019

The relation between the independence number and rank of a signed graph

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

math.CO↗

arXiv2018

The Component Connectivity of Alternating Group Graphs and Split-Stars

For an integer $\ell\geqslant 2$, the $\ell$-component connectivity of a graph $G$, denoted by $κ_{\ell}(G)$, 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) $κ_3(AG_n)=4n-10$ and $κ_4(AG_n)=6n-16$ for $n\geqslant 4$, and $κ_5(AG_n)=8n-24$ for $n\geqslant 5$; (ii) $κ_3(S_n^2)=4n-8$, $κ_4(S_n^2)=6n-14$, and $κ_5(S_n^2)=8n-20$ for $n\geqslant 4$.

cs.DM↗

arXiv2018

The $g$-good neighbour diagnosability of hierarchical cubic networks

Let $G=(V, E)$ be a connected graph, a subset $S\subseteq V(G)$ is called an $R^{g}$-vertex-cut of $G$ if $G-F$ is disconnected and any vertex in $G-F$ has at least $g$ neighbours in $G-F$. The $R^{g}$-vertex-connectivity is the size of the minimum $R^{g}$-vertex-cut and denoted by $κ^{g}(G)$. Many large-scale multiprocessor or multi-computer systems take interconnection networks as underlying topologies. Fault diagnosis is especially important to identify fault tolerability of such systems. The $g$-good-neighbor diagnosability such that every fault-free node has at least $g$ fault-free neighbors is a novel measure of diagnosability. In this paper, we show that the $g$-good-neighbor diagnosability of the hierarchical cubic networks $HCN_{n}$ under the PMC model for $1\leq g\leq n-1$ and the $MM^{*}$ model for $1\leq g\leq n-1$ is $2^{g}(n+2-g)-1$, respectively.

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