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).

scholarly journals Line graphs of complex unit gain graphs with least eigenvalue -2

2021 ◽  
Vol 37 (37) ◽  
pp. 14-30
Author(s):  
Maurizio Brunetti ◽  
Francesco Belardo
Keyword(s):  
Real Number ◽  
Adjacency Matrix ◽  
Multiplicative Group ◽  
Line Graph ◽  
Line Graphs ◽  
Signed Graphs ◽  
The Real ◽  
Least Eigenvalue ◽  
Gain Graphs ◽  
Gain Graph

Let $\mathbb T$ be the multiplicative group of complex units, and let $\mathcal L (\Phi)$ denote a line graph of a $\mathbb{T}$-gain graph $\Phi$. Similarly to what happens in the context of signed graphs, the real number $\min Spec(A(\mathcal L (\Phi))$, that is, the smallest eigenvalue of the adjacency matrix of $\mathcal L(\Phi)$, is not less than $-2$. The structural conditions on $\Phi$ ensuring that $\min Spec(A(\mathcal L (\Phi))=-2$ are identified. When such conditions are fulfilled, bases of the $-2$-eigenspace are constructed with the aid of the star complement technique.

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 Cospectral graphs with least eigenvalue at least -2

2005 ◽  
Vol 78 (92) ◽  
pp. 51-63 ◽  
Author(s):  
Dragos Cvetkovic ◽  
Mirko Lepovic
Keyword(s):  
Line Graph ◽  
Line Graphs ◽  
Cospectral Graphs ◽  
Least Eigenvalue

We study the phenomenon of cospectrality in generalized line graphs and in exceptional graphs. We survey old results from today's point o view and obtain some new results partly by the use of compute. Among.other things we show that a connected generalized line graph L(H) has an exceptional cospectral mate only if its root graph H, assuming it is itself connected has at most 9 vertices. The paper contains a description of a table of sets of cospectral graphs with least eigenvalue at least ?2 and at most 8 vertices together with some comments and theoretical explanations of the phenomena suggested by the table.

scholarly journals Old and new generalizations of line graphs

2004 ◽  
Vol 2004 (29) ◽  
pp. 1509-1521 ◽  
Author(s):  
Jay Bagga
Keyword(s):  
Graph Transformation ◽  
Line Graph ◽  
Line Graphs ◽  
Intersection Graphs ◽  
Connected Graphs ◽  
Graph Transformations

Line graphs have been studied for over seventy years. In 1932, H. Whitney showed that for connected graphs, edge-isomorphism implies isomorphism except forK3andK1,3. The line graph transformation is one of the most widely studied of all graph transformations. In its long history, the concept has been rediscovered several times, with different names such as derived graph, interchange graph, and edge-to-vertex dual. Line graphs can also be considered as intersection graphs. Several variations and generalizations of line graphs have been proposed and studied. These include the concepts of total graphs, path graphs, and others. In this brief survey we describe these and some more recent generalizations and extensions including super line graphs and triangle graphs.

scholarly journals Sets of cospectral graphs with least eigenvalue at least -2 and some related results

2004 ◽  
Vol 129 (29) ◽  
pp. 85-102 ◽  
Author(s):  
Dragos Cvetkovic ◽  
Mirko Lepovic
Keyword(s):  
Line Graph ◽  
Line Graphs ◽  
Cospectral Graphs ◽  
Least Eigenvalue

In this paper we study the phenomenon of cospectrality in generalized line graphs and in exceptional graphs. The paper contains a table of sets of Co spectral graphs with least eigenvalue at least ?2 and at most 8 vertices together with some comments and theoretical explanations of the phenomena suggested by the table. In particular, we prove that the multiplicity of the number 0 in the spectrum of a generalized line graph L(G) is at least the number of petals of the corresponding root graph G. .

scholarly journals An upper bound for the chromatic number of line graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Andrew D. King ◽  
Bruce A. Reed ◽  
Adrian R. Vetta
Keyword(s):  
Chromatic Number ◽  
Polynomial Time ◽  
Upper Bound ◽  
Maximum Degree ◽  
Line Graph ◽  
Polynomial Time Algorithm ◽  
Clique Number ◽  
Time Algorithm ◽  
Line Graphs ◽  
International Audience

International audience It was conjectured by Reed [reed98conjecture] that for any graph $G$, the graph's chromatic number $χ (G)$ is bounded above by $\lceil Δ (G) +1 + ω (G) / 2\rceil$ , where $Δ (G)$ and $ω (G)$ are the maximum degree and clique number of $G$, respectively. In this paper we prove that this bound holds if $G$ is the line graph of a multigraph. The proof yields a polynomial time algorithm that takes a line graph $G$ and produces a colouring that achieves our bound.

scholarly journals Graphs with least eigenvalue -2 attaining a convex quadratic upper bound for the stability number

2006 ◽  
Vol 133 (31) ◽  
pp. 41-55 ◽  
Author(s):  
D.M. Cardoso ◽  
D. Cvetkovic
Keyword(s):  
Quadratic Programming ◽  
Upper Bound ◽  
Convex Quadratic Programming ◽  
Line Graphs ◽  
Regular Graphs ◽  
Stability Number ◽  
Least Eigenvalue ◽  
Regularity Properties ◽  
New Construction ◽  
The Stability

In this paper we study the conditions under which the stability number of line graphs, generalized line graphs and exceptional graphs attains a convex quadratic programming upper bound. In regular graphs this bound is reduced to the well known Hoffman bound. Some vertex subsets inducing sub graphs with regularity properties are analyzed. Based on an observation concerning the Hoffman bound a new construction of regular exceptional graphs is provided. AMS Mathematics Subject Classification (2000): 05C50.

scholarly journals Topological invariants for the line graphs of some classes of graphs

2019 ◽  
Vol 17 (1) ◽  
pp. 1483-1490
Author(s):  
Xiaoqing Zhou ◽  
Mustafa Habib ◽  
Tariq Javeed Zia ◽  
Asim Naseem ◽  
Anila Hanif ◽  
...  
Keyword(s):  
Communication Theory ◽  
Chemical Reactivity ◽  
Line Graph ◽  
Chemical Properties ◽  
Topological Invariants ◽  
Line Graphs ◽  
Physical And Chemical Properties ◽  
Physical And Chemical ◽  
Ladder Graphs ◽  
And Chemical Properties

AbstractGraph theory plays important roles in the fields of electronic and electrical engineering. For example, it is critical in signal processing, networking, communication theory, and many other important topics. A topological index (TI) is a real number attached to graph networks and correlates the chemical networks with physical and chemical properties, as well as with chemical reactivity. In this paper, our aim is to compute degree-dependent TIs for the line graph of the Wheel and Ladder graphs. To perform these computations, we first computed M-polynomials and then from the M-polynomials we recovered nine degree-dependent TIs for the line graph of the Wheel and Ladder graphs.

A NOTE ON GENERALIZED RAINBOW CONNECTION OF CONNECTED GRAPHS AND THEIR NUMBER OF EDGES

2021 ◽  
Vol 66 (3) ◽  
pp. 3-7
Author(s):  
Anh Nguyen Thi Thuy ◽  
Duyen Le Thi
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Connection Number ◽  
Connected Graphs ◽  
Rainbow Connection Number ◽  
Rainbow Connection ◽  
Rainbow Path ◽  
Vertex Disjoint

Let l ≥ 1, k ≥ 1 be two integers. Given an edge-coloured connected graph G. A path P in the graph G is called l-rainbow path if each subpath of length at most l + 1 is rainbow. The graph G is called (k, l)-rainbow connected if any two vertices in G are connected by at least k pairwise internally vertex-disjoint l-rainbow paths. The smallest number of colours needed in order to make G (k, l)-rainbow connected is called the (k, l)-rainbow connection number of G and denoted by rck,l(G). In this paper, we first focus to improve the upper bound of the (1, l)-rainbow connection number depending on the size of connected graphs. Using this result, we characterize all connected graphs having the large (1, 2)-rainbow connection number. Moreover, we also determine the (1, l)-rainbow connection number in a connected graph G containing a sequence of cut-edges.

scholarly journals A Strengthening of Brooks' Theorem for Line Graphs

10.37236/632 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Landon Rabern
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Line Graph ◽  
Clique Number ◽  
Line Graphs ◽  
Delta G

We prove that if $G$ is the line graph of a multigraph, then the chromatic number $\chi(G)$ of $G$ is at most $\max\left\{\omega(G), \frac{7\Delta(G) + 10}{8}\right\}$ where $\omega(G)$ and $\Delta(G)$ are the clique number and the maximum degree of $G$, respectively. Thus Brooks' Theorem holds for line graphs of multigraphs in much stronger form. Using similar methods we then prove that if $G$ is the line graph of a multigraph with $\chi(G) \geq \Delta(G) \geq 9$, then $G$ contains a clique on $\Delta(G)$ vertices. Thus the Borodin-Kostochka Conjecture holds for line graphs of multigraphs.

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