arXiv subjects

Zenan Du

Publications and source records attributed to Zenan Du.

5 recordsLinked to original sources

arXiv2026

Signed graphs with exactly two main eigenvalues: The unicyclic case

An eigenvalue $\lambda$ of a signed graph $S$ of order $n$ is called a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector $j$. Characterizing signed graphs with exactly $k$ $(1\le k\le n)$ distinct main eigenvalues is a problem in algebraic and graph theory that has been studied since 2020. Du et al. (2024, 2026) characterized a class of signed graphs with exactly two main eigenvalues by analyzing a type of multigraph whose base graph is a tree. In this paper, we extend this study to the case where the associated multigraph has a unicyclic base graph, and we conclude by proposing several open problems.

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arXiv2025

Signless Laplacian spectral conditions for even factors in graphs

A spanning subgraph $F$ of a graph $G$ is defined as an even factor of $G$, if the degree $d_F(v)=2k, k\in\mathbb{N}^+$ for every vertex $v\in V(G)$. This note establishes a sufficient condition to ensure that a connected graph $G$ of even order with the minimum degree $\delta$ contains an even factor based on the signless Laplacian spectral radius.

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arXiv2024

Spectral conditions for spanning $k$-trees or $k$-ended-trees of $t$-connected graphs

Let $G$ be a connected graph of order $n$. A spanning $k$-tree of $G$ is a spanning tree with the maximum degree at most $k$, and a spanning $k$-ended-tree of $G$ is a spanning tree at most $k$ leaves, where $k\geq2$ is an integer. This paper establishes some spectral conditions for the existence of spanning $k$-trees or spanning $k$-ended-trees in $t$-connected graphs, which generalize the results of Fan et al. (2022) and Zhou (2010), and improve the results of Fiedler et al. (2010), Ao et al. (2023) and Ao et al. (2025).

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arXiv2023

Note on VDB Topological Indices of k-Cyclic Graphs

Let $G$ be a connected graph with $n$ vertices and $m$ edges. The vertex-degree-based topological index (VDB) (or graphical function-index) $TI(G)$ of $G$ with edge-weight function $I(x,y)$ is defined as $$TI(G)=\sum\limits_{uv\in E(G)}I(d_{u},d_{v}),$$ where $I(x,y)>0$ is a symmetric real function with $x\geq 1$ and $y\geq 1$, $d_{u}$ is the degree of vertex $u$ in $G$. In this note, we deduce a number of previously established results, and state a few new. For a VDB topological index $TI$ with the property $P^{*}$, we can obtain the minimum $k$-cyclic (chemical) graphs for $k\geq3$, $n\geq 5(k-1)$. These VDB topological indices include the Sombor index, the general Sombor index, the $p$-Sombor index, the general sum-connectivity index and so on. Thus this note extends the results of Liu et al. [H. Liu, L. You, Y. Huang, Sombor index of c-cyclic chemical graphs, MATCH Commun. Math. Comput. Chem. 90 (2023) 495-504] and Ali et al. [A. Ali, D. Dimitrov, Z. Du, F. Ishfaq, On the extremal graphs for general sum-connectivity index $(χ_α)$ with given cyclomatic number when $α>1$, Discrete Appl. Math. 257 (2019) 19-30].

math.GM↗

arXiv2022

The Sombor index and coindex of two-trees

The Sombor index of a graph $G$, introduced by Ivan Gutman, is defined as the sum of the weights $\sqrt{d_G(u)^2+d_G(v)^2}$ of all edges $uv$ of $G$, where $d_G(u)$ denotes the degree of vertex $u$ in $G$. The Sombor coindex is recently defined as $\bar{SO}(G)=\sum \limits_{uv\notin E(G)}\sqrt{d_G(u)^2+d_G(v)^2}$. In this paper, the maximum and second maximum Sombor index, the minimum and second minimum Sombor coindex in two-trees are determined.

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