On friendly index sets and product-cordial index sets of gear graphs

Let G = (V,E) be a simple connected graph. A vertex labeling of $f:V→{0,1}$ of G induces two edge labelings f⁺, $f*:E→{0,1}$ defined by $f+(xy) = f(x)+f(y)(mod2)$ and $f*(xy) = f(x)f(y)$ for each edge xy ∈ E. For i∈{0,1}, let $vf(i) = |{v∈V:f(v) = i}|$⁠, $ef+(i) = |{e∈E:f+(e) = i}|$ and $ef*(i) = |e∈E:f*(e) = i}|$⁠. A labeling f is called friendly if $|vf(1)−vf(0)|≤1$⁠. The friendly index set and the product-cordial index set of G are defined as the sets {$|ef+(0)−ef+(1)|:f$ is friendly} and {$|ef*(0)−ef*(1)|:f$ is friendly}. In this paper, we completely determine the friendly index sets and product-cordial index sets of gear graphs. We also show that the product-cordial indices of a graph can be obtained from its adjacency matrix.