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A graph G is called an odd (even) graph if for every vertex v ∈ V (G), dG(v) is odd (even). Let G be a graph of even order. Scott in 1992 proved that the vertices of every connected graph of even order can be partitioned into some odd induced forests. We denote the minimum number of odd induced subgraphs which partition V (G) by od(G). If all of the subgraphs are forests, then we denote it by odF (G). In this paper, we show that if G is a connected subcubic graph of even order or G is a connected planar graph of even order, then odF (G) ≤ 4. Moreover, we show that for every tree T of even order odF (T) ≤ 2 and for every unicyclic graph G of even order odF (G) ≤ 3. Also, we prove that if G is... 

The energy of a graph G, denoted by ε(G), is defined as the sum of the absolute values of all eigenvalues of G. It is proved in [MATCH Commun. Math. Comput. Chem. 79 (2018) 287-301] by Alawiah et al. that ε(G) ≤ 2√Δ+√(n - 2)(2m - 2Δ) for every bipartite graph G of order n, size m and maximum degree Δ. We prove the above bound for all graphs G. We also prove new types of two bounds of Koolen and Moulton given in [Adv. Appl. Math. 26 (2001) 47-52] and [Graphs Comb. 19 (2003) 131-135]. © 2023 University of Kragujevac, Faculty of Science. All rights reserved  

For a graph G the Sombor index of G is defined as (Formula Presented) , where d(u) is the degree of u in G. In the current paper, we study the structure of a graph with minimum Sombor index among all graphs with fixed order and fixed size. It is shown that in every graph with minimum Sombor index the difference between minimum and maximum degrees is at most 1. © 2022 University of Kragujevac, Faculty of Science. All rights reserved  

The energy of a graph G, E(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. Gutman et al. proved that for every cubic graph of order n, E(G)≥n. Here, we improve this result by showing that if G is a connected subcubic graph of order n≥8, then E(G)≥1.01n. Also, we prove that if G is a traceable subcubic graph of order n≥8, then E(G)>1.1n. Let G be a connected cubic graph of order n≥8, it is shown that E(G)>n+2. It was proved that if G is a connected cubic graph of order n, then E(G)≤1.65n. Also, in this paper we would like to present the best lower bound for the energy of a connected cubic graph. We introduce an infinite family of connected cubic graphs whose for...