27 research outputs found
Sharp bounds on the distance spectral radius and the distance energy of graphs
No full textAbstractThe D-eigenvalues {μ1,μ2,…,…,μp} of a graph G are the eigenvalues of its distance matrix D and form the D-spectrum of G denoted by specD(G). The greatest D-eigenvalue is called the D-spectral radius of G denoted by μ1. The D-energy ED(G) of the graph G is the sum of the absolute values of its D-eigenvalues. In this paper we obtain some lower bounds for μ1 and characterize those graphs for which these bounds are best possible. We also obtain an upperbound for ED(G) and determine those maximal D-energy graphs
The distance spectrum of corona and cluster of two graphs
No full textAbstractLet G be a connected graph with a distance matrix D. The D-eigenvalues {μ1,μ2,…,…,μp} of G are the eigenvalues of D and form the distance spectrum or D-spectrum of G. Given two graphs G with vertex set {v1,v2,……,vp} and H, the corona G∘H is defined as the graph obtained by taking p copies of H and for each i, joining the ith vertex of G to all the vertices in the ith copy of H. Let H be a rooted graph rooted at u. Then the cluster G{H} is defined as the graph obtained by taking p copies of H and for each i, joining the ith vertex of G to the root in the ith copy of H. In this paper we describe the distance spectrum of G∘H, for a connected distance regular graph G and any r-regular graph H in terms of the distance spectrum of G and adjacency spectrum of H. We also describe the distance spectrum of G{Kn}, where G is a connected distance regular graph
Some classes of Turker equivalent graphs
No full textTwo graphs G and H are Turker equivalent if they have the same set of Turker angles.
In this paper some Turker equivalent family of graphs are obtained
Distance spectrum of Indu–Bala product of graphs
No full textAbstractThe D-eigenvalues μ1,μ2,…,μn of a graph G of order n are the eigenvalues of its distance matrix D and form the distance spectrum or D-spectrum of G denoted by SpecD(G). Let G1 and G2 be two regular graphs. The Indu–Bala product of G1 and G2 is denoted by G1▾G2 and is obtained from two disjoint copies of the join G1∨G2 of G1 and G2 by joining the corresponding vertices in the two copies of G2. In this paper we obtain the distance spectrum of G1▾G2 in terms of the adjacency spectra of G1 and G2. We use this result to obtain a new class of distance equienergetic graphs of diameter 3. We also prove that the class of graphs Kn¯▾Kn+1¯ has integral distance spectrum
On the distance spectral radius and the distance energy of graphs
No full textThe D-eigenvalues {mu(1), mu(2), ... , mu(p)} of a connected graph G are the eigenvalues of its distance matrix D. The D-energy of a graph G is the sum of the absolute values of its D-eigenvalues denoted by E(D)(G). In this article, we obtain a lower bound for the largest D-eigenvalue of G and an upper bound for E(D)(G) which improve Indulal's bounds [G. Indulal, Sharp bounds on the distance spectral radius and the distance energy of graphs, Linear Algebra Appl. 430 (2009), pp. 106-113]. In the final section of the article, we give an important remark on the distance regular graphs
On graphs preserving PI index upon edge removal
No full textThe paper is concerned with the PI index of graphs. Let G be a graph and e its edge. If PI(G) = PI(G- e) , then e is said to be a PI-invariant edge of G. Bipartite graphs have no PI-invariant edges. A general class of non-bipartite graphs is constructed, possessing PI-invariant edges
On the distance spectra of some graphs
No full textThe D-eigenvalues of a connected graph G are the eigenvalues of its distance matrix D, and form the D-spectrum of G. The D-energy E_{D}(G) of the graph G is the sum of the absolute values of its
D-eigenvalues. Two (connected) graphs are said to be
D-equienergetic if they have equal D-energies. The D-spectra
of some graphs and their D-energies are calculated. A pair of
D-equienergetic bipartite graphs on , ,
vertices is constructed
ON DISTANCE ENERGY OF GRAPHS
No full textThe D-eigenvalues of a graph G are the eigenvalues of its distance matrix D, and the
D-energy ED(G) is the sum of the absolute values of its D-eigenvalues. Two graphs are
said to be D-equienergetic if they have the same D-energy. In this note we obtain bounds
for the distance spectral radius and D-energy of graphs of diameter 2. Pairs of equiregular
D-equienergetic graphs of diameter 2, on p = 3t + 1 vertices are also constructed
The distance spectrum and energy of the compositions of regular graphs
No full textAbstractThe distance energy of a graph G is a recently developed energy-type invariant, defined as the absolute deviation of the eigenvalues of the distance matrix of G. It is a useful molecular descriptor in QSPR modelling, as demonstrated by Consonni and Todeschini in [V. Consonni, R. Todeschini, New spectral indices for molecule description, MATCH Commun. Math. Comput. Chem. 60 (2008) 3–14]. We describe here the distance spectrum and energy of the join-based compositions of regular graphs in terms of their adjacency spectrum. These results are used to show that there exist a number of families of sets of noncospectral graphs with equal distance energy, such that for any n∈N, each family contains a set with at least n graphs. The simplest such family consists of sets of complete bipartite graphs
