scholarly journals Distance spectra and distance energies of iterated line graphs of regular graphs

2009 ◽  
Vol 85 (99) ◽  
pp. 39-46 ◽  
Author(s):  
H.S. Ramane ◽  
D.S. Revankar ◽  
Ivan Gutman ◽  
H.B. Walikar
Keyword(s):  
Line Graph ◽  
Distance Matrix ◽  
Line Graphs ◽  
Regular Graphs ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Iterated Line Graph ◽  
The Absolute ◽  
Equienergetic Graphs

The distance or D-eigenvalues of a graph G are the eigenvalues of its distance matrix. The distance or D-energy ED(G) of the graph G is the sum of the absolute values of its D-eigenvalues. Two graphs G1 and G2 are said to be D-equienergetic if ED(G1) = ED(G2). Let F1 be the 5-vertex path, F2 the graph obtained by identifying one vertex of a triangle with one end vertex of the 3-vertex path, F3 the graph obtained by identifying a vertex of a triangle with a vertex of another triangle and F4 be the graph obtained by identifying one end vertex of a 4-vertex star with a middle vertex of a 3-vertex path. In this paper we show that if G is r-regular, with diam(G)? 2, and Fi,i = 1,2,3,4, are not induced subgraphs of G, then the k-th iterated line graph Lk(G) has exactly one positive D-eigenvalue. Further, if G is r-regular, of order n, diam(G)?2, and G does not have Fi,i=1,2,3,4, as an induced subgraph, then for k ?1, ED(Lk(G)) depends solely on n and r. This result leads to the construction of non D-cospectral, D-equienergetic graphs having same number of vertices and same number of edges.

scholarly journals Nonplanarity of Iterated Line Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Jing Wang
Keyword(s):  
Line Graph ◽  
Crossing Number ◽  
Line Graphs ◽  
Full Characterization ◽  
Iterated Line Graph

The 1-crossing index of a graph G is the smallest integer k such that the k th iterated line graph of G has crossing number greater than 1. In this paper, we show that the 1-crossing index of a graph is either infinite or it is at most 5. Moreover, we give a full characterization of all graphs with respect to their 1-crossing index.

Reciprocal complementary distance equienergetic graphs

2016 ◽  
Vol 09 (04) ◽  
pp. 1650084 ◽  
Author(s):  
Harishchandra S. Ramane ◽  
Gouramma A. Gudodagi
Keyword(s):  
Regular Graphs ◽  
Energy Formula ◽  
The Absolute ◽  
Equienergetic Graphs

The reciprocal complementary distance (RCD) matrix of a graph [Formula: see text] is defined as [Formula: see text], in which [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text], where [Formula: see text] is the diameter of [Formula: see text] and [Formula: see text] is the distance between the [Formula: see text]th and [Formula: see text]th vertex of [Formula: see text]. The [Formula: see text]-energy [[Formula: see text]] of [Formula: see text] is defined as the sum of the absolute values of the eigenvalues of RCD-matrix of [Formula: see text]. Two graphs [Formula: see text] and [Formula: see text] are said to be RCD-equienergetic if [Formula: see text]. In this paper, we obtain the RCD-eigenvalues and RCD-energy of the join of certain regular graphs and thus construct the non-RCD-cospectral, RCD-equienergetic graphs on [Formula: see text] vertices, for all [Formula: see text].

scholarly journals Some Properties of Regular Line Graphs

2008 ◽  
pp. 44-49
Keyword(s):  
Key Words ◽  
Bipartite Graph ◽  
Line Graph ◽  
Bipartite Graphs ◽  
Line Graphs ◽  
Sufficient Condition ◽  
Regular Graphs ◽  
Necessary And Sufficient Condition ◽  
Regular Line ◽  
Necessary And Sufficient

In this paper, the concept of regular line graph has been introduced. The maximum number of vertices with different degrees in the regular line graphs has also been studied. Further, the necessary and sufficient condition for regular line graph to be bipartite graph have also been proved. Key words: Line Graphs, Regular graphs, Connected graphs, Bipartite Graphs.

scholarly journals Maximum Degree Growth of the Iterated Line Graph

10.37236/1460 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Stephen G. Hartke ◽  
Aparna W. Higgins
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Line Graph ◽  
Induced Subgraphs ◽  
Iterated Line Graph

Let $\Delta_k$ denote the maximum degree of the $k^{\rm th}$ iterated line graph $L^k(G)$. For any connected graph $G$ that is not a path, the inequality $\Delta_{k+1}\leq 2\Delta_k-2$ holds. Niepel, Knor, and Šoltés have conjectured that there exists an integer $K$ such that, for all $k\geq K$, equality holds; that is, the maximum degree $\Delta_k$ attains the greatest possible growth. We prove this conjecture using induced subgraphs of maximum degree vertices and locally maximum vertices.

Spectrum of graphs obtained by operations

2018 ◽  
Vol 13 (02) ◽  
pp. 2050045
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Somnath Paul
Keyword(s):  
Distance Matrix ◽  
Laplacian Matrix ◽  
Regular Graphs ◽  
Lexicographic Product ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian ◽  
Adjacency Spectrum ◽  
Simple Connected Graph ◽  
Equienergetic Graphs

The distance signless Laplacian matrix of a simple connected graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix whose main diagonal entries are the vertex transmissions in [Formula: see text]. In this paper, we first determine the distance signless Laplacian spectrum of the graphs obtained by generalization of the join and lexicographic product graph operations (namely joined union) in terms of their adjacency spectrum and the eigenvalues of an auxiliary matrix, determined by the graph [Formula: see text]. As an application, we show that new pairs of auxiliary equienergetic graphs can be constructed by joined union of regular graphs.

scholarly journals Seidel Equienergetic Graphs

2016 ◽  
Vol 16 ◽  
pp. 62-69
Author(s):  
Harishchandra S. Ramane ◽  
Mahadevappa M. Gundloor ◽  
Sunilkumar M. Hosamani
Keyword(s):  
Characteristic Polynomial ◽  
Regular Graphs ◽  
Seidel Matrix ◽  
The Absolute ◽  
Equienergetic Graphs ◽  
Square Matrix

The Seidel matrix S(G) of a graph G is the square matrix with diagonal entries zeroes and off diagonal entries are – 1 or 1 corresponding to the adjacency and non-adjacency. The Seidel energy SE(G) of G is defined as the sum of the absolute values of the eigenvalues of S(G). Two graphs G1 and G2 are said to be Seidel equienergetic if SE(G1) = SE(G2). We establish an expression for the characteristic polynomial of the Seidel matrix and for the Seidel energy of the join of regular graphs. Thereby construct Seidel non cospectral, Seidel equienergetic graphs on n vertices, for all n ≥ 12

Modeling Boiling Points of Cycloalkanes by Means of Iterated Line Graph Sequences

2001 ◽  
Vol 41 (4) ◽  
pp. 1041-1045 ◽  
Author(s):  
Željko Tomović ◽  
Ivan Gutman
Keyword(s):  
Line Graph ◽  
Iterated Line Graph ◽  
Boiling Points

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.

scholarly journals Spectra and energies of iterated line graphs of regular graphs

2005 ◽  
Vol 18 (6) ◽  
pp. 679-682 ◽  
Author(s):  
H.S. Ramane ◽  
H.B. Walikar ◽  
S.B. Rao ◽  
B.D. Acharya ◽  
P.R. Hampiholi ◽  
...  
Keyword(s):  
Line Graphs ◽  
Regular Graphs

scholarly journals Matrix Partitions with Finitely Many Obstructions

10.37236/976 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Tomás Feder ◽  
Pavol Hell ◽  
Wing Xie
Keyword(s):  
Symmetric Matrix ◽  
Forbidden Subgraph ◽  
Input Graph ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Forbidden Induced Subgraphs ◽  
Extra Effort ◽  
Matrix Partition ◽  
Finite Set ◽  
Forbidden Induced Subgraph

Each $m$ by $m$ symmetric matrix $M$ over $0, 1, *$, defines a partition problem, in which an input graph $G$ is to be partitioned into $m$ parts with adjacencies governed by $M$, in the sense that two distinct vertices in (possibly equal) parts $i$ and $j$ are adjacent if $M(i,j)=1$, and nonadjacent if $M(i,j)=0$. (The entry $*$ implies no restriction.) We ask which matrix partition problems admit a characterization by a finite set of forbidden induced subgraphs. We prove that matrices containing a certain two by two diagonal submatrix $S$ never have such characterizations. We then develop a recursive technique that allows us (with some extra effort) to verify that matrices without $S$ of size five or less always have a finite forbidden induced subgraph characterization. However, we exhibit a six by six matrix without $S$ which cannot be characterized by finitely many induced subgraphs. We also explore the connection between finite forbidden subgraph characterizations and related questions on the descriptive and computational complexity of matrix partition problems.

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