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2. Bounds for the (Laplacian) spectral radius of graphs with parameter $\alpha $
- Creator:
- Tian, Gui-Xian and Huang, Ting-Zhu
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- graph, adjacency matrix, Laplacian matrix, spectral radius, and bound
- Language:
- English
- Description:
- Let $G$ be a simple connected graph of order $n$ with degree sequence $(d_1,d_2,\ldots ,d_n)$. Denote $(^\alpha t)_i = \sum \nolimits _{j\colon i \sim j} {d_j^\alpha }$, $(^\alpha m)_i = {(^\alpha t)_i }/{d_i^\alpha }$ and $(^\alpha N)_i = \sum \nolimits _{j\colon i \sim j} {(^\alpha t)_j }$, where $\alpha $ is a real number. Denote by $\lambda _1(G)$ and $\mu _1(G)$ the spectral radius of the adjacency matrix and the Laplacian matrix of $G$, respectively. In this paper, we present some upper and lower bounds of $\lambda _1(G)$ and $\mu _1(G)$ in terms of $(^\alpha t)_i $, $(^\alpha m)_i $ and $(^\alpha N)_i $. Furthermore, we also characterize some extreme graphs which attain these upper bounds. These results theoretically improve and generalize some known results.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
3. Directed pseudo-graphs and Lie algebras over finite fields
- Creator:
- Boza, Luis, Fedriani, Eugenio Manuel, Núňez, Juan, Pacheco, Ana María, and Villar, María Trinidad
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- directed pseudo-graph, adjacency matrix, and Lie algebra
- Language:
- English
- Description:
- The main goal of this paper is to show an application of Graph Theory to classifying Lie algebras over finite fields. It is rooted in the representation of each Lie algebra by a certain pseudo-graph. As partial results, it is deduced that there exist, up to isomorphism, four, six, fourteen and thirty-four $2$-, $3$-, $4$-, and $5$-dimensional algebras of the studied family, respectively, over the field $\mathbb {Z}/2\mathbb {Z}$. Over $\mathbb {Z}/3\mathbb {Z}$, eight and twenty-two $2$- and $3$-dimensional Lie algebras, respectively, are also found. Finally, some ideas for future research are presented.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
4. Spectral characterization of multicone graphs
- Creator:
- Wang, Jianfeng, Zhao, Haixing, and Huang, Qiongxiang
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- adjacency matrix, cospectral graph, spectral characteriztion, and multicone graph
- Language:
- English
- Description:
- A multicone graph is defined to be the join of a clique and a regular graph. Based on Zhou and Cho's result [B. Zhou, H. H. Cho, Remarks on spectral radius and Laplacian eigenvalues of a graph, Czech. Math. J. 55 (130) (2005), 781–790], the spectral characterization of multicone graphs is investigated. Particularly, we determine a necessary and sufficient condition for two multicone graphs to be cospectral graphs and investigate the structures of graphs cospectral to a multicone graph. Additionally, lower and upper bounds for the largest eigenvalue of a multicone graph are given.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
5. Unicyclic graphs with bicyclic inverses
- Creator:
- Panda, Swarup Kumar
- Format:
- print, bez média, and svazek
- Type:
- model:article and TEXT
- Subject:
- matematika, mathematics, adjacency matrix, unicyclic graph, bicyclic graph, inverse graph, perfect matching, 13, and 51
- Language:
- English
- Description:
- A graph is nonsingular if its adjacency matrix $A(G)$ is nonsingular. The inverse of a nonsingular graph $G$ is a graph whose adjacency matrix is similar to $A(G)^{-1}$ via a particular type of similarity. Let $\mathcal{H}$ denote the class of connected bipartite graphs with unique perfect matchings. Tifenbach and Kirkland (2009) characterized the unicyclic graphs in $\mathcal{H}$ which possess unicyclic inverses. We present a characterization of unicyclic graphs in $\mathcal{H}$ which possess bicyclic inverses., Swarup Kumar Panda., and Obsahuje bibliografii
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public