Abstract
Let \(G\) be a connected graph with the vertex set \(V = \{v_1,
v_2, \ldots, v_n\}\), where \(n \geq 2\). Denote by di the degree
of the vertex \(v_i\) for \(i = 1, 2, \ldots, n\). If \(v_i\) and
\(v_j\) are adjacent in \(G\), we write \(i \sim j\), otherwise
we write \(i \nsim j\). The variable sum exdeg index and coindex
of \(G\) are defined as \(SEI_a(G) =\sum_{i\sim j} (a^{d_i} + a^{d_j})
= \sum_{i=1}^n d_ia^{d_i}\) and \(\overline{SEI}_a(G) =
\sum_{i\nsim j} (a^{d_i} + a^{d_j}) = \sum_{i=1}^n (n - 1 - d_i)a^{d_i}\),
respectively, where `\(a\)' is a positive real number different from 1.
Some inequalities involving \(SEI_a(G)\) or/and \(\overline{SEI}_a(G)\)
are derived. Special cases of the obtained inequalities are also discussed
for unicyclic graphs.
