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## Publications

Publications (35)

Let G be a simple and undirected graph with n vertices. The row entries corresponding to the vertex v in the adjacency matrix of G are denoted by s(v). The number of positions at which the elements of the strings s(u) and s(v) differ is the Hamming distance between them. The sum of Hamming distances between all the pairs of vertices is the Hamming...

The energy E(G) of a graph G, defined as the sum of the absolute values of its eigenvalues, belongs to the most popular graph invariants in chemical graph theory. It originates from the π−electron energy in the Huckel molecular orbital model, but has also gained purely mathematical interest. Let q1, q2,. .. , qn be the signless Laplacian eigenvalue...

The topological indices are the numerical parameters associated with the graph which are usually graph invariant. The topological indices are classified based on the properties of graphs. The degree distance index is the topological index which is calculated by counting the degrees and distance between the vertices. In this paper, the degree distan...

Let G be a graph with vertex set V. A set D ⊆ V is a dominating set of G if each vertex of V − D is adjacent to at least one vertex of D. The k (k(i))− complement of G is obtained by partitioning V into k partites and removing the edges between the vertices of different (same) partites in G and adding the edges between the vertices of different (sa...

Let A(G) be the adjacency matrix of a graph G. Let s(vi) denote the row entries of A(G) corresponding to the vertex vi of G. The Hamming distance between the strings s(ui) and s(vi) is the number of positions in which their elements differ. The sum of Hamming distance between all the pairs of vertices is the Hamming index of a graph. In this paper,...

The color energy of a graph G is defined as the sum of the absolute values of the color eigenvalues of G. The graphs with large number of edges are referred as cluster graphs. Cluster graphs are obtained from complete graphs by deleting few edges according to some criteria. Bipartite cluster graphs are obtained by deleting few edges from complete b...

Let [Formula: see text] be the adjacency matrix of a graph [Formula: see text]. Let [Formula: see text] denote the row entries of [Formula: see text] corresponding to the vertex [Formula: see text] of [Formula: see text]. The Hamming distance between the strings [Formula: see text] and [Formula: see text] is the number of positions in which [Formul...

Let G(V,X) be a finite and simple graph of order n and size m. The complement of G, denoted by G¯, is the graph obtained by removing the lines of G and adding the lines that are not in G. A graph is self-complementary if and only if it is isomorphic to its complement. In this paper, we define δ-complement and δ′-complement of a graph as follows. Fo...

Let [Formula: see text] be a partition of vertex set [Formula: see text] of order [Formula: see text] of a graph [Formula: see text]. The [Formula: see text]-complement of [Formula: see text] denoted by [Formula: see text] is defined as for all [Formula: see text] and [Formula: see text] in [Formula: see text], [Formula: see text], remove the edges...

A dominating set for a graph G = (V, E) is a subset D of V such that every point not in D is adjacent to at least one member of D. Let P = {P1, P2,. .. , P k } be a partition of point set V (G). For all Pi and Pj in P of order k ≥ 2, i ̸ = j, delete the lines between Pi and Pj in G and include the lines between Pi and Pj which are not in G. The res...

The energy of the graph had its genesis in 1978. It is the sum of absolute values of its eigenvalues. It originates from the π -electron energy in the Huckel molecular orbital model but has also gained purely mathematical interest. Suppose μ1,μ2,…,μn is the Laplacian eigenvalues of G. The Laplacian energy of G has recently been defined as LE(G)=∑i=...

The color energy of a graph is defined as sum of absolute color eigenvalues of graph, denoted by Ec(G). Let Gc = (V, E) be a color graph and P = {V1, V2,. .. , V k } be a partition of V of order k ≥ 1. The k-color complement {Gc} P k of Gc is defined as follows: For all Vi and Vj in P , i ̸ = j, remove the edges between Vi and Vj and add the edges...

Let $G$ be a simple connected graph. The energy of a graph $G$ is defined as sum of the absolute eigenvalues of an adjacency matrix of the graph $G$. It represents a proper generalization of a formula valid for the total $\pi$-electron energy of a conjugated hydrocarbon as calculated by the Huckel molecular orbital (HMO) method in quantum chemistry...

An assignment of distinct colors [Formula: see text] to the vertices [Formula: see text] and [Formula: see text] of a graph [Formula: see text] such that the distance between [Formula: see text] and [Formula: see text] is at most [Formula: see text] is called [Formula: see text]-distance coloring of [Formula: see text]. Suppose [Formula: see text]...

Let G = (V,E) be a simple graph with vertex set V ={v_{1}, v_{2},...,v_{n}} and edge set E ={e_{1},e_{2},...,e_{m}}. The label incidence matrix B_{l}(G) of G is the n \times m matrix whose (i, j)-entry is a if 0 labeled edge incident to 0 labeled vertex, b if 1 labeled edge incident to 1 labeled vertex, c if unlabeled edge incident to 0 or 1 labele...

The color energy of a graph is defined as sum of absolute color eigenvalues of graph, denoted by Ec(G). Let Gc = (V, E) be a color graph and P = {V1, V2,. .. , V k } be a partition of V of order k ≥ 1. The k-color complement {Gc} P k of Gc is defined as follows: For all Vi and Vj in P , i = j, remove the edges between Vi and Vj and add the edges wh...

Let G be a finite simple graph on n vertices. Let P = {V1, V2, V3,. .. , V k } be a partition of vertex set V (G) of order k ≥ 2. For all Vi and Vj in P , i = j, remove the edges between Vi and Vj in graph G and add the edges between Vi and Vj which are not in G. The graph G P k thus obtained is called the k−complement of graph G with respect to th...

Let G be finite connected simple graph. The color energy of a graph G is defined as the sum of the absolute values of color eigenvalues of G. The derived graph of a simple graph G, denoted by G* , is a graph having same vertex set of G, in which two vertices are adjacent if and only if their distance in G is two. In this paper, we establish an uppe...

Let P = {V1, V2, V3, . . . , Vk} be a partition of vertex set V (G) of order k ≥ 2. For all Vi and Vj in P, i 6= j, remove the edges between Vi and Vj in graph G and add the edges between Vi and Vj which are not in G. The graph GPk thus obtained is called the k−complement of graph G with respect to a partition P. For each set Vr in P, remove the ed...

In this paper, we introduce the new concept of color Signless Laplacian energy LEc+(G). It depends on the underlying graph G and the colors of the vertices. Moreover, we compute color signless Laplacian spectrum and the color signless Laplacian energy of families of graph with the minimum number of colors. The color signless Laplacian energy for th...

In this paper, we introduce a new concept of color Laplacian energy LEc(G). It depends on the underlying graph G and colors of the vertices. We compute color Laplacian spectrum and color Laplacian energies of families of graph with minimum number of colors. We also obtain some bounds of color Laplacian energy. The color Laplacian energy for the col...

Let G be a binary labeled graph and A l (G) = (lij) be its label adjacency matrix. For a vertex vi, we define label degree as Li = n j=1 lij. In this paper, we define label Laplacian energy LE l (G). It depends on the underlying graph G and labels of the vertices. We compute label Laplacian spectrum of families of graph. We also obtain some bounds...

Let G be a graph with vertex set V (G) and edge set X(G) and consider the set A = {0, 1}. A mapping l : V (G) −→ A is called a binary vertex labeling of G and l(v) is called the label of the vertex v under l. In this paper we introduce a new kind of graph energy for the binary labeled graph, the minimum covering label energy E cl (G). Depending on...

Let G be graph with vertex set V(G) and edge set E(G) and the set A={0,1}. A mapping is called binary vertex labeling of G and l(v) is called the label of the vertex v under l. In this paper we introduce a new kind of graph energy for the binary labeled graph, the labeled graph energy. It depends on the underlying graph G and on its binary labeling...