Spectra of quasi-corona and multiple Q-complemented-vertex join graphs

Yang Yang; Yanyan Song; Zhanjun Si; Yi Liu; Yang Yang; Yanyan Song; Zhanjun Si; Yi Liu

doi:10.3934/math.2026422

AIMS Mathematics

2026, Volume 11, Issue 4: 10205-10225. doi: 10.3934/math.2026422

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Research article

Spectra of quasi-corona and multiple Q-complemented-vertex join graphs

1.
College of Artificial Intelligence, Tianjin University of Science and Technology, Tianjin 300457, China
2.
School of Mathematics and Statistics, Qinghai Normal University, Xining 810008, China
3.
Faculty of Civil Engineering and Mechanics, Kunming University of Science and Technology, Kunming 650500, China
4.
College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin 150001, China

Received: 28 January 2026 Revised: 29 March 2026 Accepted: 07 April 2026 Published: 15 April 2026
MSC : 05C50, 05C12, 15A18

The Q-complemented graph of a graph $ G $, denoted by $ CT(G) $, was obtained by adding a new vertex for each edge $ uv \in E(G) $, where this new vertex was adjacent to all vertices of $ G $ except $ u $ and $ v $. In this paper, we introduced two new graph constructions based on $ CT(G) $, namely, the quasi-corona Q-complemented-vertex join $ G_1 \odot G_2 $ and the multiple Q-complemented-vertex join $ G_1 \bigtriangledown G_2 $. We determined the adjacency, Laplacian, and signless Laplacian spectra of these composite graphs. Using these spectral results, we constructed infinite families of $ A $-, $ L $-, and $ Q $-cospectral graphs. Moreover, we derived explicit formulas for several key invariants of the new graphs, including the number of spanning trees, the Kirchhoff index, the signless Laplacian energy-like invariant, and graph energy. Furthermore, numerical experiments were provided to illustrate the structural properties of the proposed graphs, suggesting that the multiple Q-complemented-vertex join construction may exhibit certain small-world-like characteristics.
- Laplacian spectrum,
- cospectral graph,
- Kirchhoff index,
- spanning tree,
- graph energy
Citation: Yang Yang, Yanyan Song, Zhanjun Si, Yi Liu. Spectra of quasi-corona and multiple Q-complemented-vertex join graphs[J]. AIMS Mathematics, 2026, 11(4): 10205-10225. doi: 10.3934/math.2026422

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Abstract

The Q-complemented graph of a graph $ G $, denoted by $ CT(G) $, was obtained by adding a new vertex for each edge $ uv \in E(G) $, where this new vertex was adjacent to all vertices of $ G $ except $ u $ and $ v $. In this paper, we introduced two new graph constructions based on $ CT(G) $, namely, the quasi-corona Q-complemented-vertex join $ G_1 \odot G_2 $ and the multiple Q-complemented-vertex join $ G_1 \bigtriangledown G_2 $. We determined the adjacency, Laplacian, and signless Laplacian spectra of these composite graphs. Using these spectral results, we constructed infinite families of $ A $-, $ L $-, and $ Q $-cospectral graphs. Moreover, we derived explicit formulas for several key invariants of the new graphs, including the number of spanning trees, the Kirchhoff index, the signless Laplacian energy-like invariant, and graph energy. Furthermore, numerical experiments were provided to illustrate the structural properties of the proposed graphs, suggesting that the multiple Q-complemented-vertex join construction may exhibit certain small-world-like characteristics.

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