Let G be a graph. The first Zagreb M1(G) of graph G is defined as: M1(G) = uV(G) deg(u)2. In this paper, we prove that each even number except 4 and 8 is a first Zagreb index of a caterpillar. Also, we show that the fist Zagreb index cannot be an odd number. Moreover, we obtain the fist Zagreb index of some graph operations.

Edge (5, 3)-bicolorings of a (5, 3)-regular bipartite graph relating 5cycles and vertices of the Petersen graph lead to solutions of the Great Circle Challenge puzzle. This takes for odd n = 2k + 1 > 1 to a oneto-one correspondence from the family Cn of n-cycles of Kn onto the family Ok of n-cycles of the odd graph Ok+1. It also takes for odd l with 3 ≤ l ≤ ⌊n/2⌋ to a graph having Ok as vertex ...

A graph G is a prime distance graph (respectively, a 2-odd graph) if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the difference of their labels is prime (either 2 or odd). We prove that trees, cycles, and bipartite graphs are prime distance graphs, and that Dutch windmill graphs and paper mill graphs are prime distance graphs if and only if the Tw...

Let $G$ be a graph with $p$ vertices and $q$ edges. The graph $G$ is said to be a super pair sum labeling if there exists a bijection $f$ from $V(G)cup E(G)$ to ${0, pm 1, pm2, dots, pm (frac{p+q-1}{2})}$ when $p+q$ is odd and from $V(G)cup E(G)$ to ${pm 1, pm 2, dots, pm (frac{p+q}{2})}$ when $p+q$ is even such that $f(uv)=f(u)+f(v).$ A graph that admits a super pair sum labeling is called a {...

Cavenagh and Wanless recently proved that, for sufficiently large odd n, the number of transversals in the Latin square formed from the addition table for integers modulo n is greater than (3.246). We adapt their proof to show that for sufficiently large t the number of additive permutations on [−t, t] is greater than (3.246) and we go on to derive some much improved lower bounds on the numbers...

let g be a graph. the first zagreb m1(g) of graph g is defined as: m1(g) = uv(g) deg(u)2. in this paper, we prove that each even number except 4 and 8 is a first zagreb index of a caterpillar. also, we show that the fist zagreb index cannot be an odd number. moreover, we obtain the fist zagreb index of some graph operations.