scholarly journals Least eigenvalue of the connected graphs whose complements are cacti

2019 ◽  
Vol 17 (1) ◽  
pp. 1319-1331
Author(s):  
Haiying Wang ◽  
Muhammad Javaid ◽  
Sana Akram ◽  
Muhammad Jamal ◽  
Shaohui Wang
Keyword(s):  
Linear Algebra ◽  
Adjacency Matrix ◽  
Connected Graphs ◽  
Least Eigenvalue ◽  
The Family

Abstract Suppose that Γ is a graph of order n and A(Γ) = [ai,j] is its adjacency matrix such that ai,j is equal to 1 if vi is adjacent to vj and ai,j is zero otherwise, where 1 ≤ i, j ≤ n. In a family of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix is minimum in the set of the least eigenvalues of all the graphs. Petrović et al. [On the least eigenvalue of cacti, Linear Algebra Appl., 2011, 435, 2357-2364] characterized a minimizing graph in the family of all cacti such that the complement of this minimizing graph is disconnected. In this paper, we characterize the minimizing graphs G ∈ $\begin{array}{} {\it\Omega}^c_n \end{array}$, i.e. $$\begin{array}{} \displaystyle \lambda_{min}(G)\leq\lambda_{min}(C^c) \end{array}$$ for each Cc ∈ $\begin{array}{} {\it\Omega}^c_n \end{array}$, where $\begin{array}{} {\it\Omega}^c_n \end{array}$ is a collection of connected graphs such that the complement of each graph of order n is a cactus with the condition that either its each block is only an edge or it has at least one block which is an edge and at least one block which is a cycle.

scholarly journals On the nullity of connected graphs with least eigenvalue at least -2

2013 ◽  
Vol 7 (2) ◽  
pp. 250-261 ◽  
Author(s):  
Jiang Zhou ◽  
Lizhu Sun ◽  
Hongmei Yao ◽  
Changjiang Bu
Keyword(s):  
Upper Bound ◽  
Adjacency Matrix ◽  
Line Graph ◽  
Line Graphs ◽  
Connected Graphs ◽  
Least Eigenvalue

Let L (resp. L+) be the set of connected graphs with least adjacency eigenvalue at least -2 (resp. larger than -2). The nullity of a graph G, denoted by ?(G), is the multiplicity of zero as an eigenvalue of the adjacency matrix of G. In this paper, we give the nullity set of L+ and an upper bound on the nullity of exceptional graphs. An expression for the nullity of generalized line graphs is given. For G ? L, if ?(G) is sufficiently large, then G is a proper generalized line graph (G is not a line graph).

GRAPHS WITH SMALL INDEPENDENCE NUMBER MINIMIZING THE SPECTRAL RADIUS

2013 ◽  
Vol 05 (03) ◽  
pp. 1350017 ◽  
Author(s):  
XUE DU ◽  
LINGSHENG SHI
Keyword(s):  
Spectral Radius ◽  
Linear Algebra ◽  
Adjacency Matrix ◽  
Independence Number ◽  
Maximum Size ◽  
Maximum Eigenvalue ◽  
Connected Graphs ◽  
Blowing Up ◽  
Number Formula

The independence number of a graph is defined as the maximum size of a set of pairwise non-adjacent vertices and the spectral radius is defined as the maximum eigenvalue of the adjacency matrix of the graph. Xu et al. in [The minimum spectral radius of graphs with a given independence number, Linear Algebra and its Applications431 (2009) 937–945] determined the connected graphs of order n with independence number [Formula: see text] which minimize the spectral radius. In this paper, we show that the graph obtained from a path of order α by blowing up each vertex to a clique of order k minimizes the spectral radius among all connected graphs of order kα with independence number α for α = 3, 4 and conjecture that this is true for all α ∈ ℕ.

scholarly journals Corrigendum to “The effect on eigenvalues of connected graphs by adding edges” [Linear Algebra Appl. 548 (2018) 57–65]

2018 ◽  
Vol 555 ◽  
pp. 434-436
Author(s):  
Ji-Ming Guo ◽  
Zhi-Wen Wang ◽  
Jianxi Li ◽  
Wai Chee Shiu ◽  
Pan-Pan Tong
Keyword(s):  
Linear Algebra ◽  
Connected Graphs

Identification of kinematic chains and distinct mechanisms using extended adjacency matrix

2007 ◽  
Vol 221 (1) ◽  
pp. 81-88 ◽  
Author(s):  
A Mohammad ◽  
R A Khan ◽  
V P Agrawal
Keyword(s):  
Adjacency Matrix ◽  
Kinematic Chain ◽  
Theoretic Approach ◽  
Kinematic Chains ◽  
Graph Theoretic ◽  
The Past ◽  
Graph Theoretic Approach ◽  
The Family ◽  
Eigen Value ◽  
Eigen Values

Development of the methods for generating distinct mechanisms derived from a given family of kinematic chains has been persued by a number of researchers in the past, as the distinct kinematic structures provide distinct performance characteristics. A new method is proposed to identify the distinct mechanisms derived from a given kinematic chain in this paper. Kinematic chains and their derived mechanisms are represented in the form of an extended adjacency matrix [EA] using the graph theoretic approach. Two structural invariants derived from the eigen spectrum of the [EA] matrix are the sum of absolute eigen values EA∑ and maximum absolute eigen value EAmax. These invariants are used as the composite identification number of a kinematic chain and mechanism and are tested to identify the all-distinct mechanisms derived from the family of 1-F kinematic chains up to 10 links. The identification of distinct kinematic chains and their mechanisms is necessary to select the best possible mechanism for the specified task at the conceptual stage of design.

scholarly journals Laplacian integral graphs with a given degree sequence constraint

2021 ◽  
Vol 40 (6) ◽  
pp. 1431-1448
Author(s):  
Ansderson Fernandes Novanta ◽  
Carla Silva Oliveira ◽  
Leonardo de Lima
Keyword(s):  
Adjacency Matrix ◽  
Degree Sequence ◽  
Laplacian Matrix ◽  
Bipartite Graphs ◽  
Diagonal Matrix ◽  
Connected Graphs ◽  
Sequence Constraint ◽  
The Matrix ◽  
Vertex Degrees ◽  
Integral Graphs

Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) −A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral if all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all Lintegral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.

scholarly journals The algebraic size of the family of injective operators

2017 ◽  
Vol 15 (1) ◽  
pp. 13-20 ◽  
Author(s):  
Luis Bernal-González
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Algebra ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
The Family ◽  
One To One ◽  
Algebra Of Operators

Abstract In this paper, a criterion for the existence of large linear algebras consisting, except for zero, of one-to-one operators on an infinite dimensional Banach space is provided. As a consequence, it is shown that every separable infinite dimensional Banach space supports a commutative infinitely generated free linear algebra of operators all of whose nonzero members are one-to-one. In certain cases, the assertion holds for nonseparable Banach spaces.

scholarly journals Further results on the least eigenvalue of connected graphs

2011 ◽  
Vol 435 (9) ◽  
pp. 2303-2313 ◽  
Author(s):  
Miroslav Petrović ◽  
Tatjana Aleksić ◽  
Slobodan Simić
Keyword(s):  
Connected Graphs ◽  
Least Eigenvalue

scholarly journals Balanced Centrality of Networks

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Mark Debono ◽  
Josef Lauri ◽  
Irene Sciriha
Keyword(s):  
Graph Theory ◽  
Network Analysis ◽  
Linear Combination ◽  
Linear Algebra ◽  
Adjacency Matrix ◽  
Clear Understanding ◽  
Degree Centrality ◽  
Two Dimensional ◽  
Dimensional Array

There is an age-old question in all branches of network analysis. What makes an actor in a network important, courted, or sought? Both Crossley and Bonacich contend that rather than its intrinsic wealth or value, an actor’s status lies in the structures of its interactions with other actors. Since pairwise relation data in a network can be stored in a two-dimensional array or matrix, graph theory and linear algebra lend themselves as great tools to gauge the centrality (interpreted as importance, power, or popularity, depending on the purpose of the network) of each actor. We express known and new centralities in terms of only two matrices associated with the network. We show that derivations of these expressions can be handled exclusively through the main eigenvectors (not orthogonal to the all-one vector) associated with the adjacency matrix. We also propose a centrality vector (SWIPD) which is a linear combination of the square, walk, power, and degree centrality vectors with weightings of the various centralities depending on the purpose of the network. By comparing actors’ scores for various weightings, a clear understanding of which actors are most central is obtained. Moreover, for threshold networks, the (SWIPD) measure turns out to be independent of the weightings.

scholarly journals The Orderings of Bicyclic Graphs and Connected Graphs by Algebraic Connectivity

10.37236/434 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Jianxi Li ◽  
Ji-Ming Guo ◽  
Wai Chee Shiu
Keyword(s):  
Linear Algebra ◽  
Laplacian Matrix ◽  
Algebraic Connectivity ◽  
Connected Graphs ◽  
Unicyclic Graphs ◽  
Smallest Eigenvalue ◽  
Bicyclic Graphs

The algebraic connectivity of a graph $G$ is the second smallest eigenvalue of its Laplacian matrix. Let $\mathscr{B}_n$ be the set of all bicyclic graphs of order $n$. In this paper, we determine the last four bicyclic graphs (according to their smallest algebraic connectivities) among all graphs in $\mathscr{B}_n$ when $n\geq 13$. This result, together with our previous results on trees and unicyclic graphs, can be used to further determine the last sixteen graphs among all connected graphs of order $n$. This extends the results of Shao et al. [The ordering of trees and connected graphs by their algebraic connectivity, Linear Algebra Appl. 428 (2008) 1421-1438].

scholarly journals Forbidden Pairs with a Common Graph Generating Almost the Same Sets

10.37236/6190 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Shuya Chiba ◽  
Jun Fujisawa ◽  
Michitaka Furuya ◽  
Hironobu Ikarashi
Keyword(s):  
Symmetric Difference ◽  
Forbidden Subgraph ◽  
Transitive Graph ◽  
Forbidden Subgraphs ◽  
Induced Subgraph ◽  
Connected Graphs ◽  
The Family ◽  
Vertex Transitive ◽  
Vertex Transitive Graph

Let $\mathcal{H}$ be a family of connected graphs. A graph $G$ is said to be $\mathcal{H}$-free if $G$ does not contain any members of $\mathcal{H}$ as an induced subgraph. Let $\mathcal{F}(\mathcal{H})$ be the family of connected $\mathcal{H}$-free graphs. In this context, the members of $\mathcal{H}$ are called forbidden subgraphs.In this paper, we focus on two pairs of forbidden subgraphs containing a common graph, and compare the classes of graphs satisfying each of the two forbidden subgraph conditions. Our main result is the following: Let $H_{1},H_{2},H_{3}$ be connected graphs of order at least three, and suppose that $H_{1}$ is twin-less. If the symmetric difference of $\mathcal{F}(\{H_{1},H_{2}\})$ and $\mathcal{F}(\{H_{1},H_{3}\})$ is finite and the tuple $(H_{1};H_{2},H_{3})$ is non-trivial in a sense, then $H_{2}$ and $H_{3}$ are obtained from the same vertex-transitive graph by successively replacing a vertex with a clique and joining the neighbors of the original vertex and the clique. Furthermore, we refine a result in [Combin. Probab. Comput. 22 (2013) 733–748] concerning forbidden pairs.

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