For a graph G with vertex set V(G) = {v1, v2, . . . , vn}, the extended double cover G∗ is a bipartite graph with bipartition (X, Y), X = {x1, x2, . . . , xn} and Y = {y1, y2, . . . , yn}, where two vertices xi and yj are adjacent if and only if i = j or vi adjacent to vj in G. The double graphD[G] of G is a graph obtained by taking two copies of G and joining each vertex in one copy with the n...

The D-eigenvalues {μ1, μ2, . . . , μn} of a graph G are the eigenvalues of its distance matrix D and form the D-spectrum of G denoted by specD(G) . The D-energy ED(G) of the graph G is the sum of the absolute values of its D-eigenvalues. We describe here the distance spectrum of some self-complementary graphs in the terms of their adjacency spectrum. These results are used to show that there ex...

We prove that the Cayley graphs $X(G,S)$ and $X^+(G,S)$ are equienergetic for any abelian group $G$ symmetric subset $S$. then focus on family of unitary $G_R=X(R,R^*)$, where $R$ is a finite commutative ring with identity. show under mild conditions, $\{G_R, G_R^+\}$ pairs integral non-isospectral (generically connected non-bipartite). Then, we obtain conditions such \bar G_R\}$ graphs. Finall...

In this paper, we determine the Randić energy of m-splitting graph, m-shadow graph and m-duplicate a given m being an arbitrary integer. Our results allow construction infinite sequence graphs having same energy. Further, some invariants like degree Kirchhoff index, Kemeny’s constant number spanning trees special graphs. From our results, indicate how to obtain infinitely many pairs equienerget...

The reciprocal complementary distance (RCD) matrix of a graphG is defined asRCD(G) = [rcij] where rcij = 1 1+D−dij if i 6= j and rcij = 0, otherwise, where D is the diameter of G and dij is the distance between the vertices vi and vj 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 RCDequienergetic if they have same R...