On the Extremal Zagreb Indices of Graphs with Cut Edges

Abstract

For a (molecular) graph, the first Zagreb index M 1 is equal to the sum of squares of the vertex degrees, and the second Zagreb index M 2 is equal to the sum of products of degrees of pairs of adjacent vertices. In this paper, we show that all connected graphs with n vertices and k cut edges, the maximum (resp. minimum) M 1 -and M 2 -value are obtained, respectively, and uniquely, at K k n (resp. P k n ), where K k n is a graph obtained by joining k independent vertices to one vertex of K n−k and P k n is a graph obtained by connecting a pendent path P k+1 to one vertex of C n−k .

On general reduced second Zagreb index of graphs

“…Corollary 3.3. [5,10,11] Let G be a graph in G k n . Then…”

“…The extremal graphs of order n with k cut edges on Zagreb indices were studied in [10,11]. Namely, it was proved that K k n has maximum M 1 or M 2 -value in G k n .…”

On the Maximum Zagreb Indices of Graphs with k Cut Vertices

“…Mathematicians too exhibited considerable interest in the properties of the Randić connectivity index; see [8,9,21,22,36,37,44,45,47]. The Randić connectivity index has also evolved into several variants [20,46,49,50] The research is supported in part by National Science The Randić connectivity index has been extended as the general Randić connectivity index and general zeroth-order Randić connectivity index, and then the Zagreb indices appear to be the special cases of them [11,13,14,19,25,29,37]. The Zagreb indices have been introduced in 1972 in the report of Gutman and Trinajstić on the topological basis of the π -electron energy [23]-two terms appeared in the topological formula for the total π -energy of alternant hydrocarbons, which were in 1975 used by Gutman et al [24] as branching indices, denoted by M 1 and M 2 , and later employed as molecular descriptors in QSPR and QSAR; see [4,5].…”

Sharp bounds for the Zagreb indices of bicyclic graphs with k-pendant vertices

“…An important direction is determining graphs with maximal and minimal Zagreb indices in a given class of graphs. Indeed, over a significant class of graphs, the bounds for M 1 and M 2 were obtained (see [4,6,7,10,11,14,[17][18][19][20][21][22][23][24]26,27,[29][30][31]). Chen and Deng [5] studied the zeroth-order general Randić index, i.e., w α (G) = ∑ u∈V (G) d(u) α , where α is a real number, for n-vertex bicyclic graphs.…”