scholarly journals Spectral radius and extremal graphs for class of unicyclic graph with pendant vertices

2017 ◽  
Vol 9 (7) ◽  
pp. 168781401770713 ◽  
Author(s):  
Lu Zhi ◽  
Meijin Xu ◽  
Xiujuan Liu ◽  
Xiaodong Chen ◽  
Chen Chen ◽  
...  
Keyword(s):  
Spectral Radius ◽  
Upper Bounds ◽  
Unicyclic Graph ◽  
Extremal Graphs ◽  
Unicyclic Graphs

In this article, we research on the spectral radius of extremal graphs for the unicyclic graphs with girth g mainly by the graft transformation and matching and obtain the upper bounds of the spectral radius of unicyclic graphs.

scholarly journals Nordhaus–Gaddum-Type Relations for Arithmetic-Geometric Spectral Radius and Energy

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Yajing Wang ◽  
Yubin Gao
Keyword(s):  
Graph Theory ◽  
Spectral Radius ◽  
Adjacency Matrix ◽  
Upper Bounds ◽  
Simple Graph ◽  
Spectral Graph Theory ◽  
Extremal Graphs ◽  
Spectral Graph ◽  
Vertex Set

Spectral graph theory plays an important role in engineering. Let G be a simple graph of order n with vertex set V=v1,v2,…,vn. For vi∈V, the degree of the vertex vi, denoted by di, is the number of the vertices adjacent to vi. The arithmetic-geometric adjacency matrix AagG of G is defined as the n×n matrix whose i,j entry is equal to di+dj/2didj if the vertices vi and vj are adjacent and 0 otherwise. The arithmetic-geometric spectral radius and arithmetic-geometric energy of G are the spectral radius and energy of its arithmetic-geometric adjacency matrix, respectively. In this paper, some new upper bounds on arithmetic-geometric energy are obtained. In addition, we present the Nordhaus–Gaddum-type relations for arithmetic-geometric spectral radius and arithmetic-geometric energy and characterize corresponding extremal graphs.

scholarly journals The Spectral Radius for a Class of Double-Star-Like Tree Systems with Maximal Degree 4

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Yufeng Mao ◽  
Meijin Xu ◽  
Xiaodong Chen ◽  
Yan-Jun Liu ◽  
Kai Li
Keyword(s):  
Spectral Radius ◽  
Transformation Method ◽  
Upper Bounds ◽  
Double Star ◽  
Maximal Degree ◽  
Extremal Graphs ◽  
Second Degree

We mainly study the properties of the 4-double-star-like tree, which is the generalization of star-like trees. Firstly we use graft transformation method to obtain the maximal and minimum extremal graphs of 4-double-star-like trees. Secondly, by the relations between the degree and second degree of vertices in maximal extremal graphs of 4-double-star-like trees we get the upper bounds of spectral radius of 4-double-star-like trees.

TWO SHARP UPPER BOUNDS FOR THE SIGNLESS LAPLACIAN SPECTRAL RADIUS OF GRAPHS

2011 ◽  
Vol 03 (02) ◽  
pp. 185-191 ◽  
Author(s):  
YA-HONG CHEN ◽  
RONG-YING PAN ◽  
XIAO-DONG ZHANG
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

The signless Laplacian matrix of a graph is the sum of its degree diagonal and adjacency matrices. In this paper, we present a sharp upper bound for the spectral radius of the adjacency matrix of a graph. Then this result and other known results are used to obtain two new sharp upper bounds for the signless Laplacian spectral radius. Moreover, the extremal graphs which attain an upper bound are characterized.

scholarly journals Solution to a Conjecture on the Maximum Skew-Spectral Radius of Odd-Cycle Graphs

10.37236/4919 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Xiaolin Chen ◽  
Xueliang Li ◽  
Huishu Lian
Keyword(s):  
Spectral Radius ◽  
Linear Algebra ◽  
Upper Bounds ◽  
Simple Graph ◽  
Vertex Degree ◽  
Maximum Matching ◽  
Extremal Graphs ◽  
Odd Cycle ◽  
Cycle Graph

Let $G$ be a simple graph with no even cycle, called an odd-cycle graph. Cavers et al. [Linear Algebra Appl. 436(12):4512-1829, 2012] showed that the spectral radius of $G^\sigma$ is the same for every orientation $\sigma$ of $G$, and equals the maximum matching root of $G$. They proposed a conjecture that the graphs which attain the maximum skew spectral radius among the odd-cycle graphs $G$ of order $n$ are isomorphic to the odd-cycle graph with one vertex degree $n-1$ and size $m=\lfloor 3(n-1)/2\rfloor$. By using the Kelmans transformation, we give a proof to the conjecture. Moreover, sharp upper bounds of the maximum matching roots of the odd-cycle graphs with given order $n$ and size $m$ are given and extremal graphs are characterized.

Bounds for the skew Laplacian spectral radius of oriented graphs

2019 ◽  
Vol 35 (1) ◽  
pp. 31-40 ◽  
Author(s):  
BILAL A. CHAT ◽  
◽  
HILAL A. GANIE ◽  
S. PIRZADA ◽  
◽  
...  
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Arbitrary Direction ◽  
Laplacian Spectral Radius ◽  
Underlying Graph ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

We consider the skew Laplacian matrix of a digraph −→G obtained by giving an arbitrary direction to the edges of a graph G having n vertices and m edges. We obtain an upper bound for the skew Laplacian spectral radius in terms of the adjacency and the signless Laplacian spectral radius of the underlying graph G. We also obtain upper bounds for the skew Laplacian spectral radius and skew spectral radius, in terms of various parameters associated with the structure of the digraph −→G and characterize the extremal graphs.

scholarly journals Improved Upper Bounds for the Laplacian Spectral Radius of a Graph

10.37236/522 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Tianfei Wang ◽  
Jin Yang ◽  
Bin Li
Keyword(s):  
Spectral Radius ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Laplacian Spectral Radius

In this paper, we present three improved upper bounds for the Laplacian spectral radius of graphs. Moreover, we determine all extremal graphs which achieve these upper bounds. Finally, some examples illustrate that the results are best in all known upper bounds in some sense.

scholarly journals Bounds for Generalized Distance Spectral Radius and the Entries of the Principal Eigenvector

2021 ◽  
Vol 52 (1) ◽  
pp. 69-89
Author(s):  
Hilal Ahmad ◽  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Gui-Xian Tian
Keyword(s):  
Spectral Radius ◽  
Distance Matrix ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Linear Combinations ◽  
Graph Parameters ◽  
Simple Connected Graph ◽  
Generalized Distance

For a simple connected graph $G$, the convex linear combinations $D_{\alpha}(G)$ of \ $Tr(G)$ and $D(G)$ is defined as $D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G)$, $0\leq \alpha\leq 1$. As $D_{0}(G)=D(G)$, $2D_{\frac{1}{2}}(G)=D^{Q}(G)$, $D_{1}(G)=Tr(G)$ and $D_{\alpha}(G)-D_{\beta}(G)=(\alpha-\beta)D^{L}(G)$, this matrix reduces to merging the distance spectral and distance signless Laplacian spectral theories. In this paper, we study the spectral properties of the generalized distance matrix $D_{\alpha}(G)$. We obtain some lower and upper bounds for the generalized distance spectral radius, involving different graph parameters and characterize the extremal graphs. Further, we obtain upper and lower bounds for the maximal and minimal entries of the $ p $-norm normalized Perron vector corresponding to spectral radius $ \partial(G) $ of the generalized distance matrix $D_{\alpha}(G)$ and characterize the extremal graphs.

scholarly journals On the α-spectral radius of graphs

2020 ◽  
pp. 22-22
Author(s):  
Haiyan Guo ◽  
Bo Zhou
Keyword(s):  
Spectral Radius ◽  
Spectral Properties ◽  
Maximum Degree ◽  
Domination Number ◽  
Upper Bounds ◽  
Unicyclic Graphs ◽  
The Family ◽  
The Difference ◽  
Graph Parameters ◽  
Irregular Graphs

For 0 ? ? ? 1, Nikiforov proposed to study the spectral properties of the family of matrices A?(G) = ?D(G)+(1 ? ?)A(G) of a graph G, where D(G) is the degree diagonal matrix and A(G) is the adjacency matrix of G. The ?-spectral radius of G is the largest eigenvalue of A?(G). For a graph with two pendant paths at a vertex or at two adjacent vertices, we prove results concerning the behavior of the ?-spectral radius under relocation of a pendant edge in a pendant path. We give upper bounds for the ?-spectral radius for unicyclic graphs G with maximum degree ? ? 2, connected irregular graphs with given maximum degree and some other graph parameters, and graphs with given domination number, respectively. We determine the unique tree with the second largest ?-spectral radius among trees, and the unique tree with the largest ?-spectral radius among trees with given diameter. We also determine the unique graphs so that the difference between the maximum degree and the ?-spectral radius is maximum among trees, unicyclic graphs and non-bipartite graphs, respectively.

Laplacian Energy of Digraphs and a Minimum Laplacian Energy Algorithm

2015 ◽  
Vol 26 (03) ◽  
pp. 367-380 ◽  
Author(s):  
Xingqin Qi ◽  
Edgar Fuller ◽  
Rong Luo ◽  
Guodong Guo ◽  
Cunquan Zhang
Keyword(s):  
Oriented Graph ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Spectral Graph Theory ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Spectral Graph ◽  
Laplacian Energy ◽  
Energy Formula ◽  
Matrix Formula

In spectral graph theory, the Laplacian energy of undirected graphs has been studied extensively. However, there has been little work yet for digraphs. Recently, Perera and Mizoguchi (2010) introduced the directed Laplacian matrix [Formula: see text] and directed Laplacian energy [Formula: see text] using the second spectral moment of [Formula: see text] for a digraph [Formula: see text] with [Formula: see text] vertices, where [Formula: see text] is the diagonal out-degree matrix, and [Formula: see text] with [Formula: see text] whenever there is an arc [Formula: see text] from the vertex [Formula: see text] to the vertex [Formula: see text] and 0 otherwise. They studied the directed Laplacian energies of two special families of digraphs (simple digraphs and symmetric digraphs). In this paper, we extend the study of Laplacian energy for digraphs which allow both simple and symmetric arcs. We present lower and upper bounds for the Laplacian energy for such digraphs and also characterize the extremal graphs that attain the lower and upper bounds. We also present a polynomial algorithm to find an optimal orientation of a simple undirected graph such that the resulting oriented graph has the minimum Laplacian energy among all orientations. This solves an open problem proposed by Perera and Mizoguchi at 2010.

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