Let G and H be two graphs. The join G ∨ H is the graph obtained by joining every vertex of G with every vertex of H. The corona G ○ H is the graph obtained by taking one copy of G and |V (G)| copies of H and joining the i-th vertex of G to every vertex in the i-th copy of H. The neighborhood corona G★H is the graph obtained by taking one copy of G and |V (G)| copies of H and joining the neighbors of the i-th vertex of G to every vertex in the i-th copy of H. The edge corona G ◇ H is the graph obtained by taking one copy of G and |E(G)| copies of H and joining each terminal vertex of i-th edge of G to every vertex in the i-th copy of H. Let G1, G2, G3 and G4 be regular graphs with disjoint vertex sets. In this paper we compute the spectrum of (G1 ∨ G2) ∪ (G1 ★ G3), (G1 ∨ G2) ∪ (G2 ★ G3) ∪ (G1 ★ G4), (G1 ∨G2)∪(G1 ○G3), (G1 ∨G2)∪(G2 ○G3)∪(G1 ○G4), (G1 ∨G2)∪(G1 ◇G3), (G1 ∨ G2) ∪ (G2 ◇ G3) ∪ (G1 ◇ G4), (G1 ∨ G2) ∪ (G2 ○ G3) ∪ (G1 ★ G3), (G1 ∨ G2) ∪ (G2 ○ G3) ∪ (G1 ◇ G4) and (G1 ∨ G2) ∪ (G2 ★ G3) ∪ (G1 ◇ G4). As an application, we show that there exist some new pairs of equienergetic graphs on n vertices for all n ≥ 11.
In this note we give an upper bound for λ(n), the maximum number of edges in a strongly multiplicative graph of order n, which is sharper than the upper bounds given by Beineke and Hegde [3] and Adiga, Ramaswamy and Somashekara [2], for n ≥ 28.
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Let G be a vertex colored graph. The minimum number χ(G) of colors needed for coloring of a graph G is called the chromatic number. Recently, Adiga et al. [1] have introduced the concept of color energy of a graph Ec(G) and computed the color energy of few families of graphs with χ(G) colors. In this paper we derive explicit formulas for the color energies of the unitary Cayley graph Xn, the complement of the colored unitary Cayley graph (Xn)c and some gcd-graphs.
In this note we give an upper bound for λ(n), the maximum number of edges in a strongly multiplicative graph of order n, which is sharper than the upper bound obtained by Beineke and Hegde [1].
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