arXiv subjects

Enqiang Zhu

Publications and source records attributed to Enqiang Zhu.

26 records · Page 2Linked to original sources

arXiv2022

PHEE: A phased hybrid evaluation-enhanced approach for identifying influential users in social networks

For the purpose of maximizing the spread of influence caused by a certain small number k of nodes in a social network, we are asked to find a k-subset of nodes (i.e., a seed set) with the best capacity to influence the nodes not in it. This problem of influence maximization (IM) has wide application, belongs to subset problems, and is NP-hard. To solve it, we should theoretically examine all seed sets and evaluate their influence spreads, which is time-consuming. Therefore, metaheuristic strategies are generally employed to gain a good seed set within a reasonable time. We observe that many algorithms for the IM problem only adopt a uniform mechanism in the whole solution search process, which lacks a response measure when the algorithm becomes trapped in a local optimum. To address this issue, we propose a phased hybrid evaluation-enhanced (PHEE) approach for IM, which utilizes two distinct search strategies to enhance the search of optimal solutions: a randomized range division evolutionary (RandRDE) algorithm to improve the solution quality, and a fast convergence strategy. Our approach is evaluated on 10 real-world social networks of different sizes and types. Experimental results demonstrate that our algorithm is efficient and obtains the best influence spread for all the datasets compared with three state-of-the-art algorithms, outperforms the time consuming CELF algorithm on four datasets, and performs worse than CELF on only two networks.

cs.SI↗

arXiv2021

Metric dimension and edge metric dimension of unicyclic graphs

The metric (resp. edge metric) dimension of a simple connected graph $G$, denoted by dim$(G)$ (resp. edim$(G)$), is the cardinality of a smallest vertex subset $S\subseteq V(G)$ for which every two distinct vertices (resp. edges) in $G$ have distinct distances to a vertex of $S$. It is an interesting topic to discuss the relation between dim$(G)$ and edim$(G)$ for some class of graphs $G$. In this paper, we settle two open problems on this topic for a widely studied class of graphs, called unicyclic graphs. Specifically, we introduce four classes of subgraphs to characterize the structure of a unicyclic graph whose metric (resp . edge metric) dimension is equal to the lower bound on this invariant for unicyclic graphs. Based on this, we determine the exact values of dim$(G)$ and edim$(G)$ for all unicyclic graphs $G$.

math.CO↗

arXiv2019

Notes on k-rainbow independent domination in graphs

The $k$-rainbow independent domination number of a graph $G$, denoted $γ_{\rm rik}(G)$, is the cardinality of a smallest set consisting of two vertex-disjoint independent sets $V_1$ and $V_2$ for which every vertex in $V(G)\setminus (V_1\cup V_2)$ has neighbors in both $V_1$ and $V_2$. This domination invariant was proposed by {Š}umenjak, Rall and Tepeh in (Applied Mathematics and Computation 333(15), 2018: 353-361), which allows to reduce the problem of computing the independent domination number of the generalized prism $G {\Box} K_k$ to an integer labeling problem on $G$. They proved a Nordhaus-Gaddum-type theorem: $5\leq γ_{\rm rik}(G)+γ_{\rm rik}(\overline{G})\leq n+3$ for every graph $G$ of order $n\geq 3$, where $\overline{G}$ is the complement of $G$. In this paper, we improve this result by showing that if $G$ is not isomorphic to the 5-cycle, then $5\leq γ_{\rm rik}(G)+γ_{\rm rik}(\overline{G})\leq n+2$. Moreover, we show that the problem of deciding whether a graph has a $k$-rainbow independent dominating function of a given weight is $\mathcal{NP}$-complete. Our results respond some open questions proposed by Šumenjak, et al.

math.CO↗

arXiv2018

On a Sufficient Condition for Planar Graphs of Maximum Degree 6 to be Totally 7-Colorable

A total $k$-coloring of a graph is an assignment of $k$ colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The Total Coloring Conjecture (TCC) states that every simple graph $G$ has a total ($Δ(G)+2$)-coloring, where $Δ(G)$ is the maximum degree of $G$. This conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6. It is known that every planar graph $G$ of $Δ(G) \geq 9$ or $Δ(G) \in \{7, 8\}$ with some restrictions has a total $(Δ(G) + 1)$-coloring. In particular, in [Shen and Wang, "On the 7 total colorability of planar graphs with maximum degree 6 and without 4-cycles", Graphs and Combinatorics, 25: 401-407, 2009], the authors proved that every planar graph with maximum degree 6 and without 4-cycles has a total 7-coloring. In this paper, we improve this result by showing that every diamond-free and house-free planar graph of maximum degree 6 is totally 7-colorable if every 6-vertex is not incident with two adjacent 4-cycles or not incident with three cycles of size $p,q,\ell$ for some $\{p,q,\ell\}\in \{\{3,4,4\},\{3,3,4\}\}$.

math.CO↗

arXiv2018

On Spectral Graph Embedding: A Non-Backtracking Perspective and Graph Approximation

Graph embedding has been proven to be efficient and effective in facilitating graph analysis. In this paper, we present a novel spectral framework called NOn-Backtracking Embedding (NOBE), which offers a new perspective that organizes graph data at a deep level by tracking the flow traversing on the edges with backtracking prohibited. Further, by analyzing the non-backtracking process, a technique called graph approximation is devised, which provides a channel to transform the spectral decomposition on an edge-to-edge matrix to that on a node-to-node matrix. Theoretical guarantees are provided by bounding the difference between the corresponding eigenvalues of the original graph and its graph approximation. Extensive experiments conducted on various real-world networks demonstrate the efficacy of our methods on both macroscopic and microscopic levels, including clustering and structural hole spanner detection.

cs.SI↗

arXiv2015

On uniquely 3-colorable plane graphs without prescribed adjacent faces

A graph $G$ is \emph{uniquely k-colorable} if the chromatic number of $G$ is $k$ and $G$ has only one $k$-coloring up to permutation of the colors. For a plane graph $G$, two faces $f_1$ and $f_2$ of $G$ are \emph{adjacent $(i,j)$-faces} if $d(f_1)=i$, $d(f_2)=j$ and $f_1$ and $f_2$ have a common edge, where $d(f)$ is the degree of a face $f$. In this paper, we prove that every uniquely 3-colorable plane graph has adjacent $(3,k)$-faces, where $k\leq 5$. The bound 5 for $k$ is best possible. Furthermore, we prove that there exist a class of uniquely 3-colorable plane graphs having neither adjacent $(3,i)$-faces nor adjacent $(3,j)$-faces, where $i,j\in \{3,4,5\}$ and $i \neq j$. One of our constructions implies that there exist an infinite family of edge-critical uniquely 3-colorable plane graphs with $n$ vertices and $\frac{7}{3}n-\frac{14}{3}$ edges, where $n(\geq 11)$ is odd and $n\equiv 2\pmod{3}$.

math.CO↗

arXiv2014

Tree-colorable maximal planar graphs

A tree-coloring of a maximal planar graph is a proper vertex $4$-coloring such that every bichromatic subgraph, induced by this coloring, is a tree. A maximal planar graph $G$ is tree-colorable if $G$ has a tree-coloring. In this article, we prove that a tree-colorable maximal planar graph $G$ with $δ(G)\geq 4$ contains at least four odd-vertices. Moreover, for a tree-colorable maximal planar graph of minimum degree 4 that contains exactly four odd-vertices, we show that the subgraph induced by its four odd-vertices is not a claw and contains no triangles.

math.CO↗

arXiv2013

Size of edge-critical uniquely 3-colorable planar graphs

A graph $G$ is \emph{uniquely k-colorable} if the chromatic number of $G$ is $k$ and $G$ has only one $k$-coloring up to permutation of the colors. A uniquely $k$-colorable graph $G$ is edge-critical if $G-e$ is not a uniquely $k$-colorable graph for any edge $e\in E(G)$. Mel'nikov and Steinberg [L. S. Mel'nikov, R. Steinberg, One counterexample for two conjectures on three coloring, Discrete Math. 20 (1977) 203-206] asked to find an exact upper bound for the number of edges in a edge-critical 3-colorable planar graph with $n$ vertices. In this paper, we give some properties of edge-critical uniquely 3-colorable planar graphs and prove that if $G$ is such a graph with $n(\geq6)$ vertices, then $|E(G)|\leq \frac{5}{2}n-6 $, which improves the upper bound $\frac{8}{3}n-\frac{17}{3}$ given by Matsumoto [N. Matsumoto, The size of edge-critical uniquely 3-colorable planar graphs, Electron. J. Combin. 20 (3) (2013) $\#$P49]. Furthermore, we find some edge-critical 3-colorable planar graphs which have $n(=10,12, 14)$ vertices and $\frac{5}{2}n-7$ edges.

math.CO↗