Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1703.08728

The spectral determinations of the multicone graphs Kw+mCn

Ali Zeydi Abdian
View original source ↗Read source document ↗Export metadata ↓

Abstract

The main goal of the paper is to characterize new classes of multicone graphs which are determined by both adjacency and Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. A wheel graph obtained from the join of a complete graph on a vertex with a cycle. A question about when wheel graphs are determined by their adjacency spectra is still unsolved. So, any indication about the determinations of these graphs with respect to their adjacency spectra can be an interesting and important problem. In [Y. Zhang, X. Liu, and X. Yong: Which wheel graphs are determined by their Laplacian spectra?. Comput. Math. Appl., 58 (2009) 1887{1890] and [M.-H. Liu: Some graphs determined by their (signless) Laplacian spectra. Czech. Math. J., 62, (2012) 1117{1134] it have been shown that except for, the wheel graph of order seven, all wheel graphs are determined by their Laplacian spectra and wheel graphs are determined by their signless Laplacian spectra, respectively. In this study, we present new classes of connected multicone graphs which are a natural generalization of wheel graphs and we show that these graphs are determined by their adjacency spectra as well as their Laplacian spectra. Also, we show that complement of some of these graphs are determined by their adjacency spectra. In addition, we give a necessary and sufficient condition for perfecting graphs cospectral with presented graphs in the paper. Finally, we pose two problems for further work.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ali Zeydi Abdian. 2018-02-27. The spectral determinations of the multicone graphs Kw+mCn. https://arxiv.org/abs/1703.08728

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On Perfect Divisibility of Bull-Free Graphs Without Long Paths

A graph $G$ is {\em perfectly divisible} if, for every induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B])<ω(H)$. Chudnovsky and Sivaraman [J. Graph Theory \textbf{90} (2019) 54-60] proved that every ($P_5$, bull)-free graph is perfectly divisible, while Chen and Xu [Discrete Appl. Math. \textbf{372} (2025) 298-307] proved the same for ($P_7,C_5$, bull)-free graphs. We extend these results by proving that every ($P_8,C_5$, bull)-free graph is perfectly divisible and that, letting $F$ denote the Grötzsch graph, a ($P_6$, bull)-free graph is perfectly divisible if and only if it is $F$-free.

math.CO↗
arXiv2026

Covering graphs by isometric trees

A connected subgraph of a graph is isometric if it preserves distances. Recently, graphs admitting a vertex or edge covering by a small number of isometric paths have been studied. In this paper, we consider the analogous problem for isometric trees, focusing on the treewidth of graphs admitting a vertex or edge covering by a small number of such trees. Baste, De Meyer, Giocanti, Objois, and Picavet showed that for coverings by two isometric trees, the treewidth is bounded. We show that already for three isometric trees, the treewidth can be linear in the number of vertices. On the positive side, we show that for graphs of bounded degree coverable by a small number of isometric trees, the treewidth is sublinear in the number of vertices.

math.CO↗
arXiv2026

Tree-independence number of $P_5$-free graphs with no large bicliques

The tree-independence number of a graph is the minimum, over all tree-decompositions of the graph, of the maximum size of an independent set contained in a bag. Graph classes of bounded tree-independence number have strong structural and algorithmic properties; however, the parameter can be unbounded even in quite restricted classes. In particular, the presence of an induced biclique $K_{\ell,\ell}$ forces tree-independence number at least $\ell$. This leads to the question whether large induced bicliques are the only obstruction to bounded tree-independence number in natural hereditary classes. A conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht states that for all positive integers $t$ and $\ell$, ${\{P_t,K_{\ell,\ell}\}}$-free graphs have bounded tree-independence number. We prove this conjecture for ${t=5}$ by showing that every ${\{P_5,K_{\ell,\ell}\}}$-free graph has tree-independence number at most ${4\ell-4}$. We also obtain related bounds for the weaker parameter of $α$-degeneracy and answer a question of Hilaire, Milanič, and Vasić whether tree-independence number of ${\{P_5,K_{\ell,\ell}\}}$-free graphs exceeds $\ell$ by at most an additive constant.

math.CO↗