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

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 New Results on Zagreb Energy of Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Seyed Mahmoud Sheikholeslami ◽  
Akbar Jahanbani ◽  
Rana Khoeilar
Keyword(s):  
Vertex Set ◽  
The Absolute ◽  
Equienergetic Graphs ◽  
Square Matrix

Let G be a graph with vertex set V G = v 1 , … , v n , and let d i be the degree of v i . The Zagreb matrix of G is the square matrix of order n whose i , j -entry is equal to d i + d j if the vertices v i and v j are adjacent, and zero otherwise. The Zagreb energy ZE G of G is the sum of the absolute values of the eigenvalues of the Zagreb matrix. In this paper, we determine some classes of Zagreb hyperenergetic, Zagreb borderenergetic, and Zagreb equienergetic graphs.

scholarly journals Degree Subtraction Adjacency Polynomial and Energy of Graphs obtained from Complete Graph

2020 ◽  
pp. 263-277
Author(s):  
Harishchandra S. Ramane ◽  
Hemaraddi N. Maraddi ◽  
Daneshwari Patil ◽  
Kavita Bhajantri
Keyword(s):  
Characteristic Polynomial ◽  
Complete Graph ◽  
Adjacency Matrix ◽  
Degree Of A Vertex ◽  
The Absolute ◽  
Square Matrix

The degree subtraction adjacency matrix of a graph G is a square matrix DSA(G)=[dij], in which dij=d(vi)-d(vj), if the vertices vi and vj are adjacent and dij=0, otherwise, where d(u) is the degree of a vertex u. The DSA energy of a graph is the sum of the absolute values of the eigenvalues of DSA matrix. In this paper, we obtain the characteristic polynomial of the DSA matrix of graphs obtained from the complete graph. Further we study the DSA energy of these graphs.

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 On the Seidel Energy of Certain Mesh Derived Networks

2019 ◽  
Vol 9 (2) ◽  
pp. 1905-1910
Keyword(s):  
Adjacency Matrix ◽  
Seidel Matrix ◽  
Energy Of Graph ◽  
The Absolute ◽  
Manual Calculation

The energy of graph G is defined as the sum of the absolute values of eigenvalues of the adjacency matrix A(G). The manual calculation of energy of graphs consumes several man hours. In this paper, we use MATLAB to generate the Seidel matrix and hence calculate the Seidel energy of some mesh derived networks.

On the Randić Matrix and Randić Energy

2021 ◽  
pp. 2150155
Author(s):  
H. Hatefi ◽  
H. Abdollahzadeh Ahangar ◽  
R. Khoeilar ◽  
S. M. Sheikholeslami
Keyword(s):  
Text Entry ◽  
The Absolute ◽  
Square Matrix

Let [Formula: see text] be a graph with [Formula: see text] vertices, and let [Formula: see text] be the degree of the vertex [Formula: see text] in the graph [Formula: see text]. The Randić matrix of [Formula: see text] is the square matrix of order [Formula: see text] whose [Formula: see text]-entry is equal to [Formula: see text] if the vertex [Formula: see text] and the vertex [Formula: see text] of [Formula: see text] are adjacent, and [Formula: see text] otherwise. The Randić eigenvalues of [Formula: see text] are the eigenvalues of its Randić matrix and the Randić energy of [Formula: see text] is the sum of the absolute values of its Randić eigenvalues. In this paper, we obtain some new results for the Randić eigenvalues and the Randić energy of a graph.

Linearized systems and a generalized Cayley–Hamilton theorem

2017 ◽  
Vol 16 (06) ◽  
pp. 1750120
Author(s):  
Jeffrey Lang ◽  
Daniel Newland
Keyword(s):  
Finite Field ◽  
Characteristic Polynomial ◽  
Minimum Distance ◽  
Solution Space ◽  
Arbitrary Field ◽  
Characteristic Polynomials ◽  
Systems Of Equations ◽  
Square Matrix

We study linearized systems of equations in characteristic [Formula: see text] of the form [Formula: see text] where [Formula: see text] is a square matrix and [Formula: see text]. We present algorithms for calculating their solutions and for determining the minimum distance of their solution spaces. In the case when [Formula: see text] has entries in [Formula: see text], the finite field of [Formula: see text] elements, we explore the relationships between the minimal and characteristic polynomials of [Formula: see text] and the above mentioned features of the solution space. In order to extend and generalize these findings to the case when [Formula: see text] has entries in an arbitrary field of characteristic [Formula: see text], we obtain generalizations of the characteristic polynomial of a matrix and the Cayley–Hamilton theorem to square linearized systems.

scholarly journals A Characteristic Polynomial for the Transition Probability Matrix of Correlated Random Walks on a Graph

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Takashi Komatsu ◽  
Norio Konno ◽  
Iwao Sato
Keyword(s):  
Random Walk ◽  
Zeta Function ◽  
Characteristic Polynomial ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Regular Graphs ◽  
Correlated Random Walk ◽  
Correlated Random Walks ◽  
Transition Probability Matrices ◽  
Probability Matrices

We define a correlated random walk (CRW) induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph $G$, and present a formula for the characteristic polynomial of the transition probability matrix of this CRW by using a determinant expression for the generalized weighted zeta function of $G$. As an application, we give the spectrum of the transition probability matrices for the CRWs induced from the Grover matrices of regular graphs and semiregular bipartite graphs. Furthermore, we consider another type of the CRW on a graph. 

scholarly journals Modular Orientations of Random and Quasi-Random Regular Graphs

2011 ◽  
Vol 20 (3) ◽  
pp. 321-329 ◽  
Author(s):  
NOGA ALON ◽  
PAWEŁ PRAŁAT
Keyword(s):  
Connected Graph ◽  
Regular Graphs ◽  
Fixed Integer ◽  
Connected Graphs ◽  
Absolute Value ◽  
The Absolute ◽  
The Difference

Extending an old conjecture of Tutte, Jaeger conjectured in 1988 that for any fixed integer p ≥ 1, the edges of any 4p-edge connected graph can be oriented so that the difference between the outdegree and the indegree of each vertex is divisible by 2p+1. It is known that it suffices to prove this conjecture for (4p+1)-regular, 4p-edge connected graphs. Here we show that there exists a finite p0 such that for every p > p0 the assertion of the conjecture holds for all (4p+1)-regular graphs that satisfy some mild quasi-random properties, namely, the absolute value of each of their non-trivial eigenvalues is at most c1p2/3 and the neighbourhood of each vertex contains at most c2p3/2 edges, where c1, c2 > 0 are two absolute constants. In particular, this implies that for p > p0 the assertion of the conjecture holds asymptotically almost surely for random (4p+1)-regular graphs.

Sign in / Sign up

Export Citation Format

Share Document