Chandrashekar Adiga

On Spectra Of Variants Of The Corona Of Two Graphs And Some New Equienergetic Graphs

Abstract 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.

On overpartition pairs into odd parts modulo powers of 2

Abstract In 2012, Lin (Electron. J. Combin. 19(2) (2012) #P17) investigated the 2 and 3-divisibility properties for p p ¯ o ( n ) , the number of overpartition pairs into odd parts. Using modular forms, he proved that for a fixed positive integer k , p p ¯ o ( n ) is almost always divisible by 2 k . In this paper, we prove several congruences for p p ¯ o ( n ) modulo higher powers of 2 in an elementary way.

Spectra of the extended neighborhood corona and extended corona of two graphs

In this paper we define extended corona and extended neighborhood corona of two graphs

An upper bound for maximum number of edges in a strongly multiplicative graph

G_{1}

Two generalizations of Ramanujan's continued fraction identities

and

On the mixed adjacency matrix of a mixed graph

G_{2}

Some new modular relations for the Rogers–Ramanujan type functions of order eleven with applications to partitions

, which are denoted by

Congruences for 7 and 49-regular partitions modulo powers of 7

G_{1}\bullet G_{2}

Congruences modulo 8 for \((2,\, k)\)-regular overpartitions for odd \(k > 1\)

and

On the constant term of the minimal polynomial of cos (2π/n) over Q

G_{1}\ast G_{2}