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On connected graphs of order n with girth g and nullity n − g
Linear Algebra and its Applications ( IF 1.0 ) Pub Date : 2021-07-30 , DOI: 10.1016/j.laa.2021.07.005
Qi Zhou , Dein Wong , Bit-Shun Tam

Let G be a simple graph of order n. The nullity of a graph G, denoted by η(G), is the multiplicity of 0 as an eigenvalue of its adjacency matrix. If G has at least one cycle, then the girth of G, denoted by gr(G), is the length of the shortest cycle in G. It is known that η(G) is bounded above by ngr(G)+2 if 4|gr(G) and by ngr(G) if 4gr(G). In this paper it is proved that when G is connected, η(G)=ngr(G)+2 if and only if G is a complete bipartite graph, different from a star, or a cycle of length a multiple of 4; that if G is not a complete bipartite graph or a cycle of length a multiple of 4, then η(G)ngr(G). Connected graphs of order n with girth g and nullity ng are characterized. This work also settles the problem of characterizing connected graphs with rank equal to girth and the problem of identifying all connected graphs G that contains a given nonsingular cycle as a shortest cycle and with the same rank as G.



中文翻译:

在周长为 g 且零点为 n − g 的 n 阶连通图上

G是一个简单的n阶图。图G的无效性,表示为η(G), 是作为其邻接矩阵特征值的 0 的重数。如果G至少有一个圈,则G的周长,表示为(G), 是G 中最短周期的长度。众所周知η(G) 上界为 n-(G)+2 如果 4|(G) 并由 n-(G) 如果 4(G). 本文证明当G连通时,η(G)=n-(G)+2当且仅当G是一个完整的二部图,不同于一个星形,或者一个长度为 4 倍数的循环;如果G不是完全二部图或长度为 4 的倍数的环,则η(G)n-(G). 具有周长g和零的n阶连通图n-G被表征。这项工作还解决了表征秩等于周长的连通图的问题,以及将包含给定非奇异环的所有连通图G识别为最短环且与G具有相同秩的问题。

更新日期:2021-08-13
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