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The distance spectral radius of graphs with given number of odd vertices
| Content Provider | Semantic Scholar |
|---|---|
| Author | Lin, Hongying Zhou, Bo |
| Copyright Year | 2016 |
| Abstract | The graphs with smallest, respectively largest, distance spectral radius among the connected graphs, respectively trees with a given number of odd vertices, are determined. Also, the graphs with the largest distance spectral radius among the trees with a given number of vertices of degree 3, respectively given number of vertices of degree at least 3, are determined. Finally, the graphs with the second and third largest distance spectral radius among the trees with all odd vertices are determined. 1. Introduction. Throughout this paper, we consider simple graphs. Let G be a connected graph with vertex set V (G) and edge set E(G). The distance between ver- tices u,v ∈ V (G), denoted by dG(u,v), is the length of a shortest path between them. The distance matrix of G, denoted by D(G), is the matrix D(G) = (dG(u,v))u,v∈V (G). Since D(G) is real and symmetric, its eigenvalues are real. The distance spectral radius of G, denoted by �(G), is the largest eigenvalue of D(G). Since D(G) is irreducible, we have by the Perron-Frobenius theorem that �(G) is simple, and there is a unique positive unit eigenvector x(G) of D(G) corresponding to �(G), which is called the distance Perron vector of G. The study of eigenvalues of the distance matrix of a connected graph dates back to the classical work of Graham and Pollack (5), Graham and Lovasz (4), and Edelberg et al. (2). For more details on spectra of distance matrices and especially on distance spectral radius, one may refer to the recent survey of Aouchiche and Hansen (1). |
| Starting Page | 286 |
| Ending Page | 305 |
| Page Count | 20 |
| File Format | PDF HTM / HTML |
| DOI | 10.13001/1081-3810.2877 |
| Volume Number | 31 |
| Alternate Webpage(s) | https://repository.uwyo.edu/cgi/viewcontent.cgi?article=2877&context=ela&httpsredir=1&referer= |
| Alternate Webpage(s) | http://repository.uwyo.edu/cgi/viewcontent.cgi?article=2877&context=ela |
| Alternate Webpage(s) | http://repository.uwyo.edu/cgi/viewcontent.cgi?article=2877&context=ela&filename=1&type=additional |
| Alternate Webpage(s) | https://doi.org/10.13001/1081-3810.2877 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
