Reciprocal Complementary Distance Spectra and Reciprocal Complementary Distance Energy of Line Graphs of Regular Graphs

The reciprocal complementary distance (RCD) matrix of a graph $G$ is defined as $RCD(G) = [rc_{ij}]$ where $rc_{ij} = \frac{1}{1+D-d_{ij}}$ if $i \neq j$ and $rc_{ij} = 0$, otherwise, where $D$ is the diameter of $G$ and $d_{ij}$ is the distance between the vertices $v_i$ and $v_j$ in $G$. The $RCD$-energy of $G$ is defined as the sum of the absolute values of the eigenvalues of $RCD(G)$. Two graphs are said to be $RCD$-equienergetic if they have same $RCD$-energy. In this paper we show that the line graph of certain regular graphs has exactly one positive $RCD$-eigenvalue. Further we show that $RCD$-energy of line graph of these regular graphs is solely depends on the order and regularity of $G$. This results enables to construct pairs of $RCD$-equienergetic graphs of same order and having different $RCD$-eigenvalues

Signless Laplacian polynomial for splice and link of graphs

The signless Laplacian matrix of a graph G is Q(G) = A(G) + D(G), where A(G) is the adjacency matrix and D(G) is the diagonal degree matrix of a graph G. The characteristic polynomial of the signless Laplacian matrix is called the signless Laplacian polynomial. The present work is all about the study of signless Laplacian polynomial for the splice of more than two graphs and the link of such graphs. It is noted that such a study is easier when we take into account of the vertex set partition being an equitable partition, because equitable partition of the vertex set reduces the computational steps and also the quotient matrix polynomial is a part of the polynomial of a graph. In this paper we consider the splice and links of complete graphs and of complete bipartite graphs and obtain the signless Laplacian polynomial of these using equitable partition of the vertex set.Publisher's Versio

Transmission and reciprocal transmission based topological co-indices of graphs

The transmission of a vertex u in a connected graph G is defined as the sum of the distances between u and all other vertices of a graph G. The reciprocal transmission of a vertex u in a connected graph G is defined as the sum of the reciprocal of distances between u and all other vertices of a graph G. In this paper, we introduce and study new topological co-indices based on the transmission and reciprocal transmission of a vertex, such as transmission and reciprocal transmission sum-connectivity co-indices, transmission and reciprocal transmission atom bond connectivity co-indices, transmission and reciprocal transmission geometric-arithmetic coindices, transmission and reciprocal transmission augmented Zagreb co-indices, and transmission and reciprocal transmission arithmetic-geometric co-indices. Further we obtain general formulae for some graphs.University Grants Commission, India (F.510/3/DRS-III/2016 (SAP -I))Ministry of Tribal Affairs, Govt. of India, New Delhi (2017 18-NFST-KAR-01182

Harmonic reciprocal status index and coindex of graphs

The reciprocal status of a vertex u is defined as the sum of reciprocal of the distances between u and all other vertices of a graph G. In this paper we have defined the harmonic reciprocal status index and coindex of a graph and obtained the bounds for it. Further the harmonic reciprocal status index and coindex of some graphs are obtained.The first author is thankful to the University Grants Commission (UGC), New Delhi for the support through grant under UGC-SAP DRS-III, 2016-2021: F.510/3/DRS-III /2016 (SAP-I).The second author is thankful to the Ministry of Tribal Affairs, Govt. of India, New Delhi for awarding National Fellowship for Higher Education No. 2017 18-NFST-KAR-01182.Publisher's Versio

An upper bound for difference of energies of a graph and its complement

The A-energy of a graph G, denoted by EA(G), is defined as sum of the absolute values of eigenvalues of adjacency matrix of G. Nikiforov in Nikiforov (2016) proved that EA(G¯)−EA(G)≤2μ¯1and EA(G)−EA(G¯)≤2μ1for any graph Gand posed a problem to find best possible upper bound for EA(G)−EA(G¯), where μ1and μ1¯are the largest adjacency eigenvalues of Gand its complement G¯respectively. We attempt to provide an answer by giving an improved upper bound on a class of graphs where regular graphs become particular case. As a consequence, it is proved that there is no strongly regular graph with negative eigenvalues greater than −1. The obtained results also improves some of the other existing results

General sum-connectivity index, general product-connectivity index, general Zagreb index and coindices of line graph of subdivision graphs

The general sum-connectivity index, general product-connectivity index, general Zagreb index and coindices of line graphs of subdivision graphs of tadpole graphs, wheels and ladders have been reported in the literature. In this paper, we obtain general expressions for these topological indices for the line graph of the subdivision graphs, thus generalizing the existing results

Signless Laplacian Polynomial and Characteristic Polynomial of a Graph

The signless Laplacian polynomial of a graph G is the characteristic polynomial of the matrix Q(G)=D(G)+A(G), where D(G) is the diagonal degree matrix and A(G) is the adjacency matrix of G. In this paper we express the signless Laplacian polynomial in terms of the characteristic polynomial of the induced subgraphs, and, for regular graph, the signless Laplacian polynomial is expressed in terms of the derivatives of the characteristic polynomial. Using this we obtain the characteristic polynomial of line graph and subdivision graph in terms of the characteristic polynomial of induced subgraphs