The main purpose of this paper is to define and investigate the Kirchhoff matrix, a new Kirchhoff index, the Kirchhoff energy and the Kirchhoff Estrada index of a graph. In addition, we establish upper and lower bounds for these new indexes and energy. In the final section, we point out a new possible application area for graphs by considering this new Kirchhoff matrix. Since graph theoretical studies (including graph parameters) consist of some fixed point techniques, they have been applied in the fields such as chemistry (in the meaning of atoms, molecules, energy etc.) and engineering (in the meaning of signal processing etc.), game theory, and physics. MSC: 05C12; 05C50; 05C90
The non-commuting graph associated to a group has non-central elements of the graph as vertices and two elements [Formula: see text] and [Formula: see text] do not form an edge if and only if [Formula: see text]. In this paper, we consider non-commuting graphs associated to dihedral and semidihedral groups. We investigate their metric properties such as center, periphery, eccentric graph, closure and interior. We also perform various types of metric identifications on these graphs. Moreover, we generate metric and metric-degree polynomials of these graphs.
For p, q, r, s, t ∈ Z + with rt ≤ p and st ≤ q, let G = G (p, q; r, s; t) be the bipartite graph with partite sets U = {u 1 , . . . Assume that p ≤ q, k < p, |U| = p, |V| = q and |E(G)| = pq -k. Then whether it is true thatIn this paper, we prove this conjecture for the range min v h ∈V {deg v h } ≤
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