We define a total k-labeling u of a graph G as a combination of an edge labeling u e : EðGÞ ! f1, 2, :::, k e g and a vertex labeling u v : VðGÞ ! f0, 2, :::, 2k v g, such that uðxÞ ¼ u v ðxÞ if x 2 VðGÞ and uðxÞ ¼ u e ðxÞ if x 2 EðGÞ, where k ¼ max fk e , 2k v g: The total k-labeling u is called an edge irregular reflexive k-labeling of G if every two different edges has distinct edge weights, where the edge weight is defined as the summation of the edge label itself and its two vertex labels. Thus, the smallest value of k for which the graph G has the edge irregular reflexive k-labeling is called the reflexive edge strength of G. In this paper, we study the edge irregular reflexive labeling of corona product of two paths and corona product of a path with isolated vertices. We determine the reflexive edge strength for these graphs.

<abstract><p>For a graph $ G $, we define a total $ k $-labeling $ \varphi $ is a combination of an edge labeling $ \varphi_e(x)\to\{1, 2, \ldots, k_e\} $ and a vertex labeling $ \varphi_v(x) \to \{0, 2, \ldots, 2k_v\} $, such that $ \varphi(x) = \varphi_v(x) $ if $ x\in V(G) $ and $ \varphi(x) = \varphi_e(x) $ if $ x\in E(G) $, then $ k = \, \mbox{max}\, \{k_e, 2k_v\} $. The total $ k $-labeling $ \varphi $ is an <italic>edge irregular reflexive $ k $-labeling</italic> of $ G $ if every two different edges $ xy $ and $ x^\prime y^\prime $, the edge weights are distinct. The smallest value $ k $ for which such labeling exists is called a <italic>reflexive edge strength</italic> of $ G $. In this paper, we focus on the edge irregular reflexive labeling of antiprism, convex polytopes $ \mathcal D_{n} $, $ \mathcal R_{n} $, and corona product of cycle with path. This study also leads to interesting open problems for further extension of the work.</p></abstract>

For a graph G, we define a total k-labeling ϕ as a combination of an edge labeling ϕe(x) → {1, 2, . . . , ke} and a vertex labeling ϕv(x) → {0, 2, . . . , 2kv}, such that ϕ(x) = ϕv(x) if x ∈ V (G) and ϕ(x) = ϕe(x) if x ∈ E(G), where k = max {ke, 2kv}. The total k-labeling ϕ is called an edge irregular reflexive k-labeling of G, if for every two edges xy, x0y0of G, one has wt(xy) 6=wt(x0y0), where wt(xy) = ϕv(x) + ϕe(xy) + ϕv(y). The smallest value of k for which such labeling exists is called a reflexive edge strength of G. In this paper, we study the edge irregular reflexive labeling on plane graphs and determine its reflexive edge strength.