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An edge labeling of a connected graph G = (V , E) is said to be local antimagic if it is a bijection f : E → { 1 , ... , | E | } such that for any pair of adjacent vertices x and y, f + (x) ≠ f + (y) , where the induced vertex label f + (x) = ∑ f (e) , with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χ la (G) , is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, we show the existence of infinitely many bipartite and tripartite graphs with χ l a = 2 , 3. [ABSTRACT FROM AUTHOR]

Let G be a graph with adjacency matrix A (G) and D (G) be the diagonal matrix of the degrees of G. Nikiforov (2017) defined the matrix A α (G) as A α (G) = α D (G) + (1 − α) A (G) , α ∈ [ 0 , 1 ]. The A α -spectral radius is the largest eigenvalue of A α (G). Let G m d be the set of all connected graphs with size m and diameter d. In this paper, we determine the graphs with maximal A α -spectral radius among all graphs in G m d. [ABSTRACT FROM AUTHOR]

For a simple graph G , let n (G) , η (G) , r (G) and g (G) be respectively the order, the nullity, the rank and the girth of G. It was shown by Cheng and Liu (2007) that for every graph G , η (G) ≤ { n (G) − g (G) + 2 if 4 | g (G) n (G) − g (G) if 4 ∤ g (G) . Connected graphs G with η (G) = n (G) − g (G) + 2 and n (G) − g (G) respectively have been characterized by Zhou et al. (2021). In this paper, we characterize connected graphs G with η (G) = n (G) − g (G) − 1. [ABSTRACT FROM AUTHOR]

The nullity η (G) of a (simple) graph G is the multiplicity of 0 as an eigenvalue of the adjacency matrix of G. It was shown by Ma et al. (2016) that for a connected graph G with n (G) vertices, e (G) edges and p (G) (≥ 1) pendant vertices, η (G) ≤ 2 c (G) + p (G) − 1 , where c (G) = e (G) − n (G) + θ (G) is the cyclomatic number of G , θ (G) being the number of connected components of G. In this paper connected graphs G and trees T that satisfy respectively the conditions η (G) = 2 c (G) + p (G) − 1 and η (T) = p (T) − 1 are characterized. Besides, we identify the matched and mismatched vertices in a tree T that satisfies η (T) = p (T) − 1 and characterize trees T that have a pendant vertex matched in T and satisfy η (T) = p (T) − 2. We obtain new sharp lower and upper bounds for a tree T with p (T) ≥ 3 , given in terms of p (T) and the numbers of two special kinds of internal paths in T. It is also proved that for any integer c ≥ 2 , the set of values attained by η (G) − n (G) + 2 m (G) , as G runs through all connected, leaf-free graphs G with cyclomatic number c , is { − c + 1 , − c + 2 , ... , 2 c − 3 , 2 c − 2 }. [ABSTRACT FROM AUTHOR]