Energy, Laplacian energy of double graphs and new families of equienergetic graphs

Abstract

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 graph D[G] of G is a graph obtained by taking two copies of G and joining each vertex in one copy with the neighbors of corresponding vertex in another copy. In this paper we study energy and Laplacian energy of the graphs G* and D[G], L-spectra of Gk* the k-th iterated extended double cover of G. We obtain a formula for the number of spanning trees of G*. We also obtain some new families of equienergetic and L-equienergetic graphs.

Bounds for graph energy in terms of vertex covering and clique numbers

“…Adding (14) and (16) and using the fact that ν + + ν − = r and ω = 2 for a bipartite graph G, we obtain…”

“…Among the pioneering results of the theory of graph energy are the lower and upper bounds for energy, see [2,5,15,16,18,19,22,26] and the references therein. For more information about energy of graph see [1,9,10,11,12,14,23] and related results see [1,24,25].…”

On Laplacian energy in terms of graph invariants

“…For its basic mathematical properties, including various lower and upper bounds, see [18,30,31,36,37] and especially the most recent works [4][5][6][7][12][13][14]19,35]. It is worth noting that LE found applications not only in theoretical organic chemistry (see, e.g., [16,24,29]), but also in image processing [32] and information theory [21].…”

Construction of L-equienergetic graphs using some graph operations

“…where two vertices x i and y j are adjacent if and only if i ¼ j or v i adjacent v j in G. It is easy to see that G Ã is connected if and only if G is connected and a vertex v i is of degree d i in G if and only if it is of degree d i þ 1 in G Ã : Following are some results associated with G Ã Proposition 1.5. [8] Let G(n, m) be a graph with L-spectra and Q-spectra as…”