Let G be a graph of minimum degree at least 1. Denote by dithe degree of a vertex viin G. The notation i not similar to j is used to indicate that the vertices viand vj are not adjacent in G. A graph of maximum degree at most 4 is known as a molecular graph. A connected graph having the same order and size is called a unicyclic graph. The symmetric division deg coindex of G is defined as SDD(G) =Pi not similar to j;vi not equal vj(d2i+d2j)(didj)-1. In this paper, new bounds for the coindex SDD(G) as well as relations between SDD(G) and some other topological indices/coindices are obtained. From one of the obtained results, it follows that the star graph (cycle graph, respectively) uniquely minimizes symmetric division deg coindex among all trees ((molecular) unicyclic graphs, respectively) of a given order.

Let G = (V, E), V = {v(1), v(2), ... , v(n)}, E = {e(1), e(2), ... , e(m)}, be a simple connected graph with n > 2 vertices and m edges, with vertex degree sequence triangle = d(1) > d(2) > > d(n) = delta, d(i)= d(v(i)), and edge degree sequence triangle(e) = d(e(1)) > d(e(2)) > > d(e(n)) = delta(e). The reformulated Sombor index is defined as RS(G) = Sigma(ei similar to ej) root d(e(i))(2 )+ d(e(j))(2). We consider a relationship between reformulated Sombor index and some of the vertex-degree-based indices.

The variable sum exdeg index and coindex of a graph G are denoted by SEIa(G) and ?SEIa(G), respectively, and they are defined as SEIa(G) = ?n,i=1 diadi and SEIa(G) = ?n,i =1(n?1?di)adi, respectively, where ?a? is a positive real number different from 1 and (d1, d2,..., dn) is the vertex-degree sequence of G. The present paper gives several new inequalities involving the graph invariants SEIa and/or ?SEIa. All graphs attaining the equality signs in the obtained inequalities are also characterized.

Let G = (V, E), V = {v1, v2, . . . , vn}, be a simple graph of order n and size m, without isolated vertices. The Sombor coindex of a graph G is defined as SO(G) = & sigma;� d2 i + d2 i �j j , where di = d(vi) is a degree of vertex vi, i = 1, 2, . . . , n. In this paper we investigate a relationship between Sombor coindex and a number of other topological coindices.

Let G be a graph of order n with eigenvalues lambda 1 >= lambda (2) > >= lambda (n) . The energy of G is defined as E (G) G ) = Sigma(n)(i=1) | lambda( i) |. In the present paper, new bounds on E ( G ) are provided. In addition, some bounds of E ( G ) are compared.

Let G be a simple graph of order n ≥ 2 with m edges. Denote by d1 ≥ d2 ≥ · · · ≥ dn > 0 the sequence of vertex degrees and by μ1 ≥ μ2 ≥ · · · ≥ μn−1 > μn = 0 the Laplacian eigenvalues of the graph G. Lower bounds for the Kirchhoff index, Kf(G) = n Σ −1 i=1 1 μi, are obtained.

Let G be a simple connected graph with n vertices. Denote by L+(G) = D(G)(-1/2)Q(G)D(G)(-1/2 )the normalized signless Laplacian matrix of graph G,where Q(G)and D(G)are the signless Laplacian and diagonal degree matrices of G,respectively. The eigenvalues of matrix L+(G),2 =gamma(+)(1)>=gamma(+)(2)>= >=gamma(+)(n)>= 0, are normalized signless Laplacian eigenvalues of G. In this paper, we introduce the normalized signless Laplacian resolvent energy of G as ERNS(G) = & sum;(n)(i=1)1/3-gamma(+)(i). We also obtain some lower and upper bounds forERNS(G)as well asits relationships with other energies and signless Kemeny's constant

Let G be a simple graph with vertex set V={v_1,v_2,… ,v_n} . The notion i∼ j is used to indicate that the vertices v_i and v_j of G are adjacent. For a vertex v_i∈ V , let d_i be the degree of v_i . The harmonic-arithmetic (HA) index of G is defined as HA(G) =∑ _i∼ j 4d_id_j(d_i+d_j)^-2 . In this paper, a considerable number of inequalities involving the HA index and other topological indices are derived. For every obtained inequality, all the graphs that satisfy the equality case are also characterized.

Let G be a non-trivial connected graph. If the vertices vi and vj of G are adjacent, we write i∼j, otherwise we write i≁j. Denote by di the degree of the vertex vi of G. The modified second Zagreb index and coindex of G are defined as M2∗(G)=∑i∼j1didj and M2∗¯(G)=∑i≁j1didj, respectively. Several lower and upper bounds on the modified second Zagreb index/coindex are derived and all the graphs attaining these bounds are characterized. The obtained bounds can be used to deduce bounds for various other topological indices.

Let G be a graph with the vertex set V = {v(1), ..., v(n)}. We use the notation i similar to j (respectively, i similar to j) for indicating that the vertices v(i) and v(j) are adjacent (respectively, non-adjacent). Denote by d(k) the degree of the vertex v(k). Most of the well-known vertex-degree-based topological indices and coindices can be represented in the forms TI(G) = Sigma(i similar to j) F(d(i),d(j)) and (TI) over bar (G) = Sigma(i not similar to j) F(d(i),d(j)), respectively, where F is a real symmetric function depending on di and dj. In this paper, several novel inequalities between some well-known topological indices/coindices are presented. The graphs for which the obtained inequalities become equalities are also characterized.

Let G be a simple connected graph of order n with m edges. Denote by gamma 1+ > gamma 2+ > center dot center dot center dot > gamma n+ > 0 the normalized signless Laplacian eigenvalues of G. In this work, we define the normalized signless Laplacian Estrada index of G as NSEE (G) = sigma ni=1 e gamma i+. Some lower bounds on NSEE (G) are also established.

Let G be a connected graph having vertex set {v1, ... ,vn} and vertex-degree sequence (d1, ... ,dn), where di represents the degree of the vertex vi . If the vertices vi and vj are adjacent in G, we write i similar to j. The arithmetic-geometric index and the geometric-arithmetic index of G are defined as AG(G) = n-ary sumation i similar to j[(di +dj)/(2Vdidj)] and GA(G) = n-ary sumation i similar to j[2Vdidj/(di +dj)], respectively. Since AG(G) and GA(G) are closely related quantities, we derive bounds on their addition as well as on their difference, namely on irrAG(G) = AG(G) - GA(G) and r(G) = AG(G) + GA(G). Some new bounds on AG(G) are also obtained.

For a tree T of order n with Laplacian eigenvalues μ1≥μ2≥⋯≥μn−1>μn=0, the Wiener index and the modified hyper-Wiener index are defined as WT=n∑i=1n−11μi and WWWT=n∑i<j1μiμj,respectively. The modified hyper-Wiener index of T is also expressed in terms of WT and Laplacian eigenvalues as WWWT=WT22n−n2∑i=1n−11μi2.In the present paper, we establish some lower and upper bounds for WWW(T).

Let G be a graph of order n. Denote by A the adjacency matrix of G and by D = diag(d(1), . . . , d(n)) the diagonal matrix of vertex degrees of G. The Laplacian matrix of G is defined as L = D - A. Let mu 1, mu 2, . . . , mu(n-1), mu(n) be eigenvalues of L satisfying mu 1 >= mu 2 >= . . .>= mu(n-1) >= mu(n) = 0. The Laplacian-energy-like invariant is a graph invariant defined as LEL(G) = Sigma(i=1) (n) (1) root mu(i). Improved upper bounds for LEL(G) are obtained and compared when G has a tree structure.

For a graph G , its atom -bond connectivity (ABC) index (respectively, atom -bond sum connectivity (ABS) index) is defined as the addition of the numbers root d(i) +d(j) - 2(d(i)d(j))(-1/2) (respectively, root d(i) +d(j) - 2(d(i)d(j))(-1/2) over all unordered pairs of adjacent vertices {v(i) ,v(j)} of G , where di and dj denote the degrees of v(i) and v(j) , respectively. In this paper, sharp upper bounds on the ABC and ABS indices are derived. All the graphs that attain the obtained bounds are also completely characterized.

Let G = (V, E), V = {v1, v2,..., vn}, be a simple connected graph of order n and size m. Denote by ?+1 ? ?+2 ?...? ?+n ? 0 the normalized signless Laplacian eigenvalues of G, and by ??(G) the sum of ?-th powers of the normalized signless Laplacian eigenvalues of a connected graph. The paper deals with bounds of ??. Some special cases, when ? = 12 and ? = ?1, are also considered.

Let G = (V, E), V = {v1, v2, . . . , vn}, be a simple graph of order n and size m, with vertex–degree sequence ∆ = d1 ≥ d2 ≥ · · · ≥ dn = δ > 0. and let s1 ≥ s2 ≥ · · · ≥ sn, si = di − δ + 1 and c1 ≤ c2 ≤ · · · ≤ cn, ci = ∆ − di + 1, be two vertex–degree like sequences. By analogy with the inverse degree graph invariant, ID(G) = ∑n i=1 1 di , the delta inverse degree and reverse inverse degree indices are defined, respectively, as δID(G) = ∑n i=1 1 si and RID(G) = ∑n i=1 1 ci . In this paper we determine sharp bounds on δID(G) and RID(G) and the extremal graphs are characterized.

Let G be a simple connected graph with n vertices. The Kirchhoff index of G is defined as Kf (G) = n E-i=1(n=1) 1/mu(i), where mu(1) >= mu(2) >= center dot center dot center dot >= mu(n-1) > mu(n) = 0 are the Laplacian eigenvalues of G. Some bounds on Kf (G) in terms of graph parameters such as the number of vertices, the number of edges, first Zagreb index, forgotten topological index, etc., are presented. These bounds improve some previously known bounds in the literature.

For a connected graph [Formula: see text] of order [Formula: see text] with normalized signless Laplacian eigenvalues [Formula: see text], the normalized signless Laplacian resolvent energy of [Formula: see text] is defined as [Formula: see text]. In this paper, we derive new inequalities on [Formula: see text] as well as relations on [Formula: see text] with Randić (normalized) incidence energy. We also deduced that some of our results improve the existing results in the literature.

Let G = (V, E) be a simple connected graph with n ≥ 2 vertices and m edges. In the literature four vertex–degree–based topological indices are associated to topological index, known as the first Zagreb index, based on the modification of a sequence of degrees. In this paper we study relationships between these topological indices and establish some new bounds.