scholarly journals The Spectral Radius of Subgraphs of Regular Graphs

10.37236/1021 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Vladimir Nikiforov
Keyword(s):  
Spectral Radius ◽  
Regular Graph ◽  
Adjacency Matrix ◽  
Regular Graphs ◽  
Proper Subgraph

Let $\mu\left( G\right) $ and $\mu_{\min}\left( G\right) $ be the largest and smallest eigenvalues of the adjacency matrix of a graph $G$. Our main results are: (i) Let $G$ be a regular graph of order $n$ and finite diameter $D.$ If $H$ is a proper subgraph of $G,$ then $$ \mu\left( G\right) -\mu\left( H\right) >{1\over nD}. $$ (ii) If $G$ is a regular nonbipartite graph of order $n$ and finite diameter $D$, then $$ \mu\left( G\right) +\mu_{\min}\left( G\right) >{1\over nD}. $$

scholarly journals The Spectral Radius and the Maximum Degree of Irregular Graphs

10.37236/956 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Sebastian M. Cioabă
Keyword(s):  
Spectral Radius ◽  
Regular Graph ◽  
Adjacency Matrix ◽  
Maximum Degree ◽  
Largest Eigenvalue ◽  
Irregular Graphs ◽  
The Largest Eigenvalue

Let $G$ be an irregular graph on $n$ vertices with maximum degree $\Delta$ and diameter $D$. We show that $$ \Delta-\lambda_1>{1\over nD}, $$ where $\lambda_1$ is the largest eigenvalue of the adjacency matrix of $G$. We also study the effect of adding or removing few edges on the spectral radius of a regular graph.

On the Largest Distance (Signless Laplacian) Eigenvalue of Non-transmission-regular Graphs

2018 ◽  
Vol 34 ◽  
pp. 459-471 ◽  
Author(s):  
Shuting Liu ◽  
Jinlong Shu ◽  
Jie Xue
Keyword(s):  
Spectral Radius ◽  
Regular Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Wiener Index ◽  
Regular Graphs ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Sum Of Distances

Let $G=(V(G),E(G))$ be a $k$-connected graph with $n$ vertices and $m$ edges. Let $D(G)$ be the distance matrix of $G$. Suppose $\lambda_1(D)\geq \cdots \geq \lambda_n(D)$ are the $D$-eigenvalues of $G$. The transmission of $v_i \in V(G)$, denoted by $Tr_G(v_i)$ is defined to be the sum of distances from $v_i$ to all other vertices of $G$, i.e., the row sum $D_{i}(G)$ of $D(G)$ indexed by vertex $v_i$ and suppose that $D_1(G)\geq \cdots \geq D_n(G)$. The $Wiener~ index$ of $G$ denoted by $W(G)$ is given by $W(G)=\frac{1}{2}\sum_{i=1}^{n}D_i(G)$. Let $Tr(G)$ be the $n\times n$ diagonal matrix with its $(i,i)$-entry equal to $TrG(v_i)$. The distance signless Laplacian matrix of $G$ is defined as $D^Q(G)=Tr(G)+D(G)$ and its spectral radius is denoted by $\rho_1(D^Q(G))$ or $\rho_1$. A connected graph $G$ is said to be $t$-transmission-regular if $Tr_G(v_i) =t$ for every vertex $v_i\in V(G)$, otherwise, non-transmission-regular. In this paper, we respectively estimate $D_1(G)-\lambda_1(G)$ and $2D_1(G)-\rho_1(G)$ for a $k$-connected non-transmission-regular graph in different ways and compare these obtained results. And we conjecture that $D_1(G)-\lambda_1(G)>\frac{1}{n+1}$. Moreover, we show that the conjecture is valid for trees.

scholarly journals Tight Estimates for Eigenvalues of Regular Graphs

10.37236/1850 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
A. Nilli
Keyword(s):  
Regular Graph ◽  
Adjacency Matrix ◽  
Regular Graphs

It is shown that if a $d$-regular graph contains $s$ vertices so that the distance between any pair is at least $4k$, then its adjacency matrix has at least $s$ eigenvalues which are at least $2 \sqrt {d-1} \cos \big({\pi\over 2 k}\big)$. A similar result has been proved by Friedman using more sophisticated tools.

Study on adjacent spectrum of two kinds of joins of graphs

2020 ◽  
Vol 34 (16) ◽  
pp. 2050179
Author(s):  
Yongbo Hou ◽  
Meifeng Dai ◽  
Changxi Dai ◽  
Tingting Ju ◽  
Yu Sun ◽  
...  
Keyword(s):  
Characteristic Polynomial ◽  
Regular Graph ◽  
Adjacency Matrix ◽  
Regular Graphs ◽  
Subdivision Graph ◽  
Analytic Expression

The multiple subdivision graph of a graph [Formula: see text], denoted by [Formula: see text], is the graph obtained by inserting [Formula: see text] paths of length 2 replacing every edge of [Formula: see text]. When [Formula: see text], [Formula: see text] is the subdivision graph of [Formula: see text]. Let [Formula: see text] be a graph with [Formula: see text] vertices and [Formula: see text] edges, [Formula: see text] be a graph with [Formula: see text] vertices and [Formula: see text] edges. The quasi-corona SG-vertex join [Formula: see text] of [Formula: see text] and [Formula: see text] is the graph obtained from [Formula: see text] and [Formula: see text] copies of [Formula: see text] by joining every vertex of [Formula: see text] to every vertex of [Formula: see text], and multiple SG-vertex join [Formula: see text] is the graph obtained from [Formula: see text] and [Formula: see text] by joining every vertex of [Formula: see text] to every vertex of [Formula: see text]. In this paper, we calculate analytic expression of characteristic polynomial of adjacency matrix of the above two types of joins of graphs for the case of [Formula: see text] being a regular graph. Then we obtain their adjacency spectra for the case of [Formula: see text] and [Formula: see text] being regular graphs.

scholarly journals Regular Factors of Regular Graphs from Eigenvalues

10.37236/431 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Hongliang Lu
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Regular Graph ◽  
Maximum Degree ◽  
Upper Bounds ◽  
Regular Graphs ◽  
Largest Eigenvalue ◽  
The Third ◽  
K Factor ◽  
Second Largest Eigenvalue

Let $r$ and $m$ be two integers such that $r\geq m$. Let $H$ be a graph with order $|H|$, size $e$ and maximum degree $r$ such that $2e\geq |H|r-m$. We find a best lower bound on spectral radius of graph $H$ in terms of $m$ and $r$. Let $G$ be a connected $r$-regular graph of order $|G|$ and $ k < r$ be an integer. Using the previous results, we find some best upper bounds (in terms of $r$ and $k$) on the third largest eigenvalue that is sufficient to guarantee that $G$ has a $k$-factor when $k|G|$ is even. Moreover, we find a best bound on the second largest eigenvalue that is sufficient to guarantee that $G$ is $k$-critical when $k|G|$ is odd. Our results extend the work of Cioabă, Gregory and Haemers [J. Combin. Theory Ser. B, 1999] who obtained such results for 1-factors.

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

scholarly journals Cycle partitions of regular graphs

2020 ◽  
pp. 1-24
Author(s):  
Vytautas Gruslys ◽  
Shoham Letzter
Keyword(s):  
Regular Graph ◽  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Simple Construction ◽  
Vertex Set ◽  
Almost All ◽  
Vertex Disjoint

Abstract Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

The maximum number of vertices of primitive regular graphs of orders 2, 3, 4 with exponent 2

2021 ◽  
pp. 97-104
Author(s):  
M. B. Abrosimov ◽  
◽  
S. V. Kostin ◽  
I. V. Los ◽  
◽  
...  
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Similar Question

In 2015, the results were obtained for the maximum number of vertices nk in regular graphs of a given order k with a diameter 2: n2 = 5, n3 = 10, n4 = 15. In this paper, we investigate a similar question about the largest number of vertices npk in a primitive regular graph of order k with exponent 2. All primitive regular graphs with exponent 2, except for the complete one, also have diameter d = 2. The following values were obtained for primitive regular graphs with exponent 2: np2 = 3, np3 = 4, np4 = 11.

Minimal Regular Graphs of Girths Eight and Twelve

1966 ◽  
Vol 18 ◽  
pp. 1091-1094 ◽  
Author(s):  
Clark T. Benson
Keyword(s):  
Group Theory ◽  
Lower Bound ◽  
Regular Graph ◽  
Regular Graphs ◽  
Image Position

In (3) Tutte showed that the order of a regular graph of degree d and even girth g > 4 is greater than or equal toHere the girth of a graph is the length of the shortest circuit. It was shown in (2) that this lower bound cannot be attained for regular graphs of degree > 2 for g ≠ 6, 8, or 12. When this lower bound is attained, the graph is called minimal. In a group-theoretic setting a similar situation arose and it was noticed by Gleason that minimal regular graphs of girth 12 could be constructed from certain groups. Here we construct these graphs making only incidental use of group theory. Also we give what is believed to be an easier construction of minimal regular graphs of girth 8 than is given in (2). These results are contained in the following two theorems.

scholarly journals SPEKTRUM PADA GRAF REGULER KUAT

2013 ◽  
Vol 5 (1) ◽  
pp. 13
Author(s):  
Rizki Mulyani ◽  
Triyani Triyani ◽  
Niken Larasati
Keyword(s):  
Regular Graph ◽  
Adjacency Matrix ◽  
Strongly Regular Graph ◽  
Simple Graph ◽  
Eigen Value ◽  
Strongly Regular

This article studied spectrum of strongly regular graph. This spectrum can be determined by the number of walk with lenght l on connected simple graph, equation of square adjacency matrix and eigen value of strongly regular graph.

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