Extremal problems for graphical function-indices and f -weighted adjacency matrix∗

Abstract Let f(x, y) (f(x)) be a symmetric real function (real function) and G = (V,E) be a graph. Denote by di the degree of a vertex i in G. The graphical function-index TIf (G) (Hf (G)) of G with edge-weight (vertex-weight) function f(x, y) (f(x)) is defined as TIf (G) = ∑ uv∈E f(du, dv) (Hf (G) = ∑ u∈V f(du)). We can also get a weighted adjacency matrix from the edge-weighted graph, i.e., Af (G) = (afij) where a f ij = f(di, dj) if vertices i and j are adjacent in G, and 0 otherwise. This matrix is simply referred to as the f -weighted adjacency matrix. One can see that the concepts of graphical function-indices and f -weighted adjacency matrix can cover all the degree-based graphical indices and degree-based adjacency matrices of graphs, such as the Zagreb indices, Randić index, ABC-index, etc., and the Randić matrix, ABC-matrix, GA-matrix, etc. So, for the graphical function-indices TIf (G) and Hf (G) and the f -weighted adjacency matrix Af (G) of a graph G, one can think about finding unified ways to study the extremal problems and spectral problems. This survey is intended to sum up the results done so far on these problems.


Notation and terminology
First, we give the necessary notation and terminology which will be used throughout this paper. We only consider simple and finite graphs in this paper. For notation and terminology not defined here, we refer the reader to [4,45]. We use V (G) and E(G) to denote the vertex-set and edge-set of a graph G, respectively. A graph G is called k-regular if the degree d(v) = k for every v ∈ V (G). Denote the degree of a vertex v in G also by d v . An (n, m)-graph is a graph G = (V (G), E(G)), where m = |E(G)| and n = |V (G)|. Let G(n, m) represent the collection of all (n, m)-graphs, G(n) represent the collection of all graphs with n vertices but without isolated vertices. We denote by ∆ and δ the maximum degree and minimum degree of G, respectively. A graph G is called almost regular if ∆ − δ ≤ 1 and G(n, m) is the family of all almost regular (n, m)-graphs. Note that for an (n, m)-graph G, if we let k = 2m/n and r = 2m − kn ∈ {0, 1, . . . , n − 1}, then G belongs to G(n, m) if and only if G has r vertices of degree k and n − r vertices of degree k + 1.
For k graphs G 1 , G 2 , . . . , G k , the union G 1 ∪ G 2 ∪ · · · ∪ G k is the graph with vertex-set V (G 1 ) ∪ V (G 2 ) ∪ · · · ∪ V (G k ) and edge-set E(G 1 ) ∪ E(G 2 ) ∪ · · · ∪ E(G k ). In particular, if G 1 = G 2 = · · · = G k = G, we denote G 1 ∪ G 2 ∪ · · · ∪ G k by kG. The join G ∨ H of two graphs G and H is the graph obtained by joining edges between each vertex of G to all vertices of H. We use C n , P n and S n to denote a cycle, a path and a star on n vertices, respectively. S d,n−d represents a double star on n vertices with degrees of the two centers equal to d and n − d, respectively.

Background
Nowadays, chemical graph theory has become a popular and important subject of study. In the mathematics subject classification 2020, "chemical graph theory" is given a subject number 05C92 in "05C Graph theory". Also, "graphical indices (Wiener index, Zagreb index, Randić index, etc.)" is given an independent subject number 05C09. The popular book [45] should be considered as one of the pioneer works on chemical graph theory.
There are many chemical indices [42], such as the first Zagreb index [40], and the Wiener index W = {u,v}⊂V (G) dist(u, v), where dist(u, v) is the distance between u and v. Usually, these chemical indices are classified into some categories: degree-based, distance-based, subgraph counting-based, eigenvalue-based, mixed, etc. In this survey, we are only concerned with the degree-based indices. We list some of the degree-based indices in Table 1; for more such indices, we refer to [18,20,33]. In the past years, a lot of work have been done on these degree-based indices; see [2,3,7,14,17,21,22,24,25,28,29,34,41,49] and the references therein.
Reciprocal sum-connectivity index Augmented Zagreb index Product-connectivity Gourava index Table 1, one can see that these chemical indices have the following unified definition. Let f (x, y) be a symmetric real function and G be a graph. The degree-based topological index or chemical index can be defined as which was mentioned by Gutman in [19]. This was also called the bond-incident-degree index BID(G) by Vukičević and Durdević in [46]. We also address it as the graphical function-index of a graph with edge-weight function f (x, y). This topological index can also be represented as where m ij is the number of edges between a vertex of degree i and a vertex of degree j and K = {(i, j) ∈ N : 1 ≤ i ≤ j ≤ n − 1}. For a family G of graphs, we call a graph G minimal in G if T I f (G) = min H∈G T I f (H), and maximal in G if T I f (G) = max H∈G T I f (H).
Another graphical function-index introduced by Linial and Rozenman in [36] is defined as follows. Let f (x) be a real function. The graphical function Notice that by taking the symmetric real function f (x, y) equal to f (x)/x + f (y)/y for some real function f (x), one can deduce that H f (G) is a special case of T I f (G).
One can see that from a graphical index, one gets a number from an edge-weighted or vertex-weighted graph, i.e., each index maps a molecular graph to a single number, which hopefully represents the structure of the graph. We often get the conclusion that for any graphical index φ, it is almost sure (with high probability) that for every graph G there is another graph G such that φ(G ) = φ(G) but G is not isomorphic to G. This is a sad news for people trying to invent graphical indices that can distinguish among graph structures. Nevertheless, if we use a matrix to represent the structure of a molecular graph with weights separately distributed on its pairs of adjacent vertices, it will completely keep the structural information of the graph, i.e., a matrix keeps much more structural information than an index. This motivates us to introduce the following notion of matrix. Let f (x, y) be a symmetric real function, and denote by d i the degree of a vertex i in G. Then a weighted adjacency matrix A f (G) of G is defined in [12] as follows: So, we get a symmetric real matrix, and call it the function-weighted adjacency matrix of G, or simply f -weighted adjacency matrix, with edge-weight function f (x, y). All the eigenvalues of A f (G) are real, say λ 1 , λ 2 , · · · , λ n and the spectral radius is ρ(A f (G)) = max 1≤i≤n |λ i |.
Although the study on the graphical function-indices T I f and H f , and on the f -weighted adjacency matrix A f has started only in recent years, there have been quite a few publications about them. So, after the above preparations, we are going to survey the results that we know in this direction. Though H f can be reduced to a T I f , we give separate surveys to them because H f has its own advantages due to the real function f (x) with only a single variable.
is a strictly convex function having property that f (x) is differentiable and its derivative is strictly convex, and G ∈ G(n, m) is connected, then it holds that Figure 1).

with equality if and only if
As one can see, Tomescu's results are all about the upper bounds of H f (G). Ali et al. in [1] gave the following lower bound for connected (n, m)-graphs with fewer edges.
Let f be a strictly convex function. Recently, we proved in [27] that the extremal graphs are exactly the almost regular graphs, which covers the result of Ali et al.

Theorem 3.5. [27]
Let n ≥ 2 and G be an (n, m)-graph with 1 ≤ m ≤ n(n − 1)/2, and let k = 2m/n and r = 2m − kn. If f is a strictly convex function, then it holds that and the equality holds if and only if G ∈ G(n, m).
Note that graphs in G(n, m) can be disconnected (see Figure 2). We then constructed connected graphs to show the following result. Figure 2: Some connected and disconnected almost regular graphs when m = n − 1.
Theorem 3.6. [27] Let n ≥ 2 and G be a connected (n, m)-graph with n − 1 ≤ m ≤ n(n − 1)/2, and let k = 2m/n and r = 2m − kn. If f is a strictly convex function, then it holds that and the equality holds if and only if G is connected and G ∈ G(n, m).
Actually, in [27] we also elaborated some algorithms to generate all the minimal graphs and minimal connected graphs, respectively. It is easy to see that G(n, m) = ∅, that is, there always exists a graph G with degree sequence In fact, it is easy to see that the degree sequence is graphical simply by verifying the conditions in [23]. At first, we give an algorithm to construct some connected graphs with this degree sequence. Algorithm 1 Find an (n, m)-graph G with degree sequence d = (d 1 , d 2 , . . . , d n ) where d i = k + 1 and d j = k for 1 ≤ i ≤ r and r + 1 ≤ j ≤ n. Input: , . . . , d (l) n ). 5: else go to 7. 6: Set l = l + 1 and go to 2. 7: return G = (V (l) , E (l−1) ).
By choosing different permutations σ in Algorithm 1, we can obtain some connected (n, m)-graphs G ∈ G(n, m) which minimize the value of H f (G) for every pairs of n and m. However, from [38] we can get the following algorithm, which can generate all graphs of G(n, m). 1: Construct a complete n-partite graph H = (P 1 , P 2 , . . . , P n ), such that each P i for 1 ≤ i ≤ r has k + 1 vertices and each P j for r + 1 ≤ j ≤ n has k vertices.    Note that to check that G s G for any G ∈ G(n, m) is a very hard nut to crack. Although this algorithm can be used to generate all graphs of G(n, m), it cannot guarantee the existence of any graph in G(n, m).
The following Algorithm 3 (similar to Algorithm 2) can be used to find all connected graphs in G(n, m).

Algorithm 3
Find all connected (n, m)-graphs with degree sequence d = (d 1 , d 2 , . . . , d n ) where d i = k + 1 and d j = k for 1 ≤ i ≤ r and r + 1 ≤ j ≤ n.
Input: n, m and d = (d 1 , d 2 , . . . , d n ) where d i = k + 1 and d j = k for 1 ≤ i ≤ r and r + 1 ≤ j ≤ n. Output: All connected graphs in G(n, m) for any given n and m, denoted by G * (n, m).
1: Construct a complete n-partite graph H = (P 1 , P 2 , . . . , P n ), such that each P i for 1 ≤ i ≤ r has k + 1 vertices and each P j for r + 1 ≤ j ≤ n has k vertices. Construct a new graph G s with vertex-set {p 1 , p 2 , . . . , p n } and p i ∼ p j if and only if there is an edge between P i and P j in M s . 6: if G s is connected with no multiple edges and G s G for any G ∈ G * (n, m) then 7: Set G * (n, m) = G * (n, m) {G s }. 8: else G * (n, m) = G * (n, m).
Note that although Algorithm 3 can be used to generate all connected graphs of G(n, m), it cannot guarantee the existence of any connected graph in G(n, m).

Results on the graphical function index T I f (G)
Rada and Cruz in [39] obtained some extremal results for graphs in G(n) with n vertices but without isolated vertices. For a topological index T I f with an associated function g(i, j) = ijf (i,j) i+j , define the sets K min (g) = {(r, s) ∈ K : g(r, s) = min g(i, j)} and K max (g) = {(r, s) ∈ K : g(r, s) = max g(i, j)}, and the complements of K min (g) and K max (g) in K are denoted by K c min (g) and K c max (g), respectively. Their results are stated as follows.
i+j . Then for every G ∈ G(n), n min

Moreover, equality on the left-hand side occurs if and only
if m pq = 0 for all (p, q) ∈ K c min (g), while equality on the right-hand side occurs if and only if m rs = 0 for all (r, s) ∈ K c max (g).
for all graphs G ∈ G(n), and equality occurs if and only if G = n−3 2 K 2 ∪ P 3 (see Figure 4).
3. If g(1, 2) = g(2, 2) ≥ g(i, j) for all (i, j) ∈ K such that (i, j) / ∈ {(1, 1), (1, 2), (2, 2)}, then for all graphs G ∈ G(n), and equality occurs if and only if G = n−3 Later, Cruz and Rada in [9] obtained some extremal results for trees with the weight function being an exponential type. Let T n be the set of all trees on n vertices. It is well known that P n and S n are extremal graphs with respect to most of graphical indices over T n , as we can see in Table 2 from [5,13,15,16,34,35,47,48,51,52].
minimal tree P n P n S n S n S n S n ? S n maximal tree S n S n P n P n P n P n S n S n/2−1 , n/2−1 (n ≥ 19) The exponential of T I f is defined as the edge-weighted topological index e T I f induced by The results of exponential topological indices from [8,9,11,50] are summarized in Table 3. Table 3: Extremal trees for some graphical function-indices with weight-functions being exponential types.
e M1 e M2 e χ e H e GA e SC e ABC e AZ minimal tree P n P n S n S n S n S n ? S n maximal tree S n S n/2−1 , n/2−1 P n P n P n P n S n S n/2−1 , n/2−1 One can see that the above know results are only about graphs on n vertices but without given number m of edges, or about functions f (x, y) of exponential type. Recently, we studied graphs with n vertices and m edges, and with much more general functions f (x, y). Before proceeding to summarizing our results, we need the following notation and terminology about the edge-weight function f (x, y). Let f 1 (x, y) = f (x + 1, y) − f (x, y) and f 2 (x, y) = f (x, y + 1) − f (x, y). Naturally, let f 11 := (f 1 ) 1 and f 12 := (f 1 ) 2 . f (x, y) is called (strictly) monotonically increasing if f 1 is non-negative (no-positive). Notice that if f (x, y) is partial differentiable and ∂f ∂x is positive (non-negative), then f 1 is positive (non-negative). The convexity of a real function is stated as follows: f (x, y) is called convex if for any (x 1 , y 1 ), (x 2 , y 2 ) and µ ∈ (0, 1), y 2 ). Notice that the convexity of f implies that f 11 ≥ 0. The following properties of a function will frequently appear. We say that a function f (x, y) has the property P (P ) if for any x 1 + y 1 = x 2 + y 2 and . It is not difficult to see that a symmetric and convex function has the property P .
Recently, we studied the extremal problems in [26] for T I f in graph family G(n, m) consisting of graphs with n vertices and m edges, and got the following results.

Theorem 4.4. [26]
Let n and m be integers such that n ≥ 2 and 1 ≤ m ≤ n(n − 1)/2. If f (x, y) is convex and partial differentiable with ∂f ∂x ≥ 0, then we have T I f (G) ≥ mf (2m/n, 2m/n) for any (n, m)-graph G, and the bound is sharp since all regular graphs can achieve the lower bound.
For graphs with very few edges, we got the following results.  mf (1, 1), and the equality holds if and only if G = mK 2 ∪ (n − 2m)K 1 (see Figure 5).
Notice that these minimal graphs are all almost regular graphs. Next, for graphs with a little more edges, we have the following result.  Theorem 4.6. [26] Suppose n and m are integers such that n ≥ 2 and n/2 ≤ m ≤ n − 1. If f (x, y) satisfies that f 1 > 0, f 11 > 0, f 12 ≥ 0 and f (1, 3) > f (2, 2), then every minimal graph in G(n, m) is an almost regular graph. Moreover, if f (1, 1) + f (2, 2) = 2f (1,2), then every almost regular graphs is also a minimal graph in G(n, m).
In this situation, the maximal graphs have the following property. When m ≤ n − 1, the unique maximal graph is the union of a star and some isolated vertices. When m is larger, the unique maximal graph among all connected (n, m)-graphs with fewer edges is shown as follows.

Extremal spectral radius results on the f -weighted adjacency matrix A f
The spectral properties, such as spectral radius, maximum and minimum eigenvalues, energy, etc. should be studied for these f -weighted adjacency matrices. Actually, there have already been many publications along with this topic. Here we would like to point out that the f -weighted adjacency matrices of graphs with degree-based weights behave quite different from the traditional (0,1)-adjacency matrix, for example, there is no monotonicity for the spectral radius, for example, if the weight function f is taken from the ABC-index. We have obtained the following results on the extremal spectral radius problem. The equalities hold if and only if T ∼ = P n or S n , respectively.

Theorem 5.3. [6]
Let f (x, y) = (x + y − 2)/(xy) be the weight function for ABC index. Then The equalities hold if and only if T ∼ = P n or S n , respectively.
One can see that the extremal graphs vary quite sensitively depending on the weight functions f . Our purpose is to study the extremal spectral radius problem in a unified way. So far we have only got the following results.
Theorem 5.4. [33] Assume that f (x, y) > 0 is a symmetric real function, increasing and convex in variable x. Then the tree of order n with largest spectral radius of A f (T ) is S n or a double star S d,n−d for some d ∈ {2, . . . , n − 2} (see Figure 7). It can be seen here that we get new extremal trees i.e., the double stars, which are different from those given in the first three results.
Theorem 5.5. [33] Assume that f (x, y) has a form P (x, y) or P (x, y), where P (x, y) is a symmetric polynomial with nonnegative coefficients and zero constant term. Then the tree of order n (n ≥ 9) with the smallest spectral radius of A f (T ) is unique and it is P n . Remark 5.1. Our approach is just a start. More efforts are needed in order to deepen the research on the extremal problems. For examples, one can focus on considering different graph or tree families. One can also consider the weight function f having some other nice properties.

Concluding remarks
It is just a starting point to unify the solutions for extremal problems of graphical function-indices and f -weighted matrix. In this way, we do not need to deal with chemical or graphical indices and degree-based weighted adjacency matrices of graphs with weight functions as various topological indices one by one separately. However, at the moment, we only solved the case when the indices are defined by a symmetric function f (x, y) with some particular properties. For those functions f (x, y) with less restrictions, further study is needed, depending on their functional properties, and on the graph families we focus on. This survey gives a hope that in the near future more results can be worked out for much wider classes of graphical function-indices and f -weighted adjacency matrices and/or for much wider families of graphs.
For graph energies, the asymptotic values were obtained in [30][31][32] for f -weighted adjacency and Laplacian matrices with degree based and degree-distance-based entries where the function f has very relaxed conditions; for example, continuous and/or differentiable. Since they are about extremal results, we omit their details in this survey.