Distribution eigenvalues and temperature index of graphs

Mohammad Reza Oboudi

doi:10.1515/spma-2025-0045

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Distribution eigenvalues and temperature index of graphs

Mohammad Reza Oboudi

Veröffentlicht/Copyright: 9. Dezember 2025

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Abstract

Let G be a simple graph on n vertices with degree sequence d 1 , … , d n . Fajtlowicz (On conjectures of Graffiti, Discrete Math. 72 (1988), 113–118) defined the temperature of a vertex v of G as d n − d , where d is the degree of v . Motivated by this definition, we define the temperature index of G , denoted by T ( G ) , as T ( G ) = d 1 n − d 1 + ⋯ + d n n − d n . We obtain some lower bounds and upper bounds for T ( G ) in terms of the number of vertices, the number of edges, the maximum and the minimum vertex degree and the Zagreb index ( Z ( G ) = d 1 2 + ⋯ + d n 2 ). Using these results we derive new bounds for the Zagreb index of graphs. Finally, we study the temperature index of graphs from the point of view of spectra of graphs (the eigenvalues of their adjacency matrices). In particular, we show that G has at least one eigenvalue in the interval [ − n − δ T ( G ) − 2 m n , n − δ T ( G ) − 2 m n ] .

Keywords: temperature index; eigenvalues of graphs; spectral radius; Zagreb index

MSC 2020: 05C31; 05C50; 15A18

1 Introduction

In this work, we consider only simple graphs. In other words, all graphs are assumed to be finite, undirected and without loops and multiple edges. Let G = ( V ( G ) , E ( G ) ) be a simple graph. The order of G denotes the number of vertices of G . For two vertices u and v by e = u v we mean the edge e between the vertices u and v . For every vertex v ∈ V ( G ) , the degree of v is the number of edges incident with v and is denoted by deg G ( v ) . The minimum degree and the maximum degree of vertices of G are denoted by δ ( G ) and Δ ( G ) , respectively. A regular graph is a graph such that every vertex of that has the same degree. The complement of G , denoted by G ¯ , is the simple graph with vertex set V ( G ) such that two distinct vertices of G ¯ are adjacent if and only if they are not adjacent in G . Let G and H be two disjoint graphs. By G ∪ H , that is called the disjoint union of G and H , we mean the graph with vertex set V ( G ) ∪ V ( H ) and edge set E ( G ) ∪ E ( H ) . In particular, by r G we mean the disjoint union of r copies of G . The edgeless graph (empty graph), the complete graph, the cycle, and the path of order n are denoted by K n ¯ , K n , C n , and P n , respectively. Let t and n 1 , … , n t be some positive integers. By K n 1 , … , n t we mean the complete multipartite graph (complete t-partite graph) with part sizes n 1 , … , n t . In particular, the complete bipartite graph with part sizes m and n is denoted by K m , n .

Let G be a simple graph with vertex set { v 1 , … , v n } . The adjacency matrix of G , denoted by A ( G ) , is the n × n matrix such that the ( i , j ) -entry is 1 if v i and v j are adjacent, and otherwise is zero. Since the adjacency matrix is real and symmetric matrix, all of its eigenvalues are real. By the eigenvalues of G we mean those of its adjacency matrix. We denote the eigenvalues of G by λ 1 ( G ) ≥ ⋯ ≥ λ n ( G ) . By the spectral radius of G, denoted by λ max ( G ) , we mean the largest eigenvalue of the adjacency matrix of G . Refer [2,7,11,12] for more details.

There are a number of graph invariants based on eigenvalues of graph correlation matrices, vertex degree sequences of graphs, and so on that have various applications in other fields such as chemistry. There is an existing movement to study graphs using invariants derived from the degree sequence and the adjacency matrix. Several of these indices have proven useful in applications (e.g., in mathematical chemistry to predict physico-chemical properties) as well as in graph theory itself (e.g., several of the older indices are present in Fajtlowicz’s computer-assisted graffiti conjectures and the later literature inspired by these). Some well-known are Randić index [13], Estrada index [3], and energy of graphs [5,9,10]. This study contributes to this movement by introducing a new invariant (the temperature index T) based on one of the earlier invariants (the degree temperature). Note that this invariant has been mentioned as a candidate by Kulli when he listed a dozen possible invariants derived from temperature, but he did not study it carefully or derive any of its properties. In 1988, Fajtlowicz [4] using Graffiti posed some conjectures related to graph invariants that deal with the degree sequence of graphs. He defined the temperature of a vertex v of G as d n − d , where d is the degree of v . Motivated by this definition, in this study, we define the temperature index of G as follows. Let G be a graph of order n and with vertex degree sequence d 1 , … , d n . We denote the temperature index of G by T ( G ) and define it as

T ( G ) = d 1 n − d 1 + ⋯ + d n n − d n .

For example, for every n ≥ 3 , T ( P n ) = 2 + 2 n − 1 and T ( C n ) = 2 n n − 2 and for every n ≥ 1 , T ( K n ) = n ( n − 1 ) . One of the interesting examples is the temperature index of complete bipartite graphs. One can easily check that T ( K p , q ) = p + q .

The Zagreb index (the first Zagreb index) of G was defined as Z ( G ) = d 1 2 + ⋯ + d n 2 [6]. In Section 2, we obtain some lower bounds and upper bounds for T ( G ) in terms of the number of vertices, the number of edges, the maximum and the minimum vertex degree, and the Zagreb index. In particular, we show that

2 m n + Z ( G ) n 2 − n δ ( G ) ≤ T ( G ) ≤ 2 m n + Z ( G ) n 2 − n Δ ( G ) .

Using these results, we derive new bounds for the Zagreb index of graphs. In Section 3, we study the temperature index of graphs based on the spectra of graphs, that is, in point of view of the eigenvalues of their adjacency matrices. We show that if m is the number of edges of G , then G has at least one eigenvalue in the interval − n − δ T − 2 m n , n − δ T − 2 m n , where T = T ( G ) and δ = δ ( G ) . We obtain that λ ( G ) ≥ ( n − Δ ) ( T − 2 m n ) and the equality holds if and only if G is regular, where λ ( G ) is the spectral radius of G , Δ = Δ ( G ) , and T = T ( G ) .

2 Temperature index of graphs

In this section, we obtain some bounds for the temperature index of graphs in terms of the number of vertices and the number of edges of graphs.

Theorem 1

Let G be a graph with n vertices and m edges. Then,

2 m n ≤ T ( G ) ≤ 2 m .

In addition, in the left-hand side, the equality holds if and only if G = K n ¯ and in the right-hand side, the equality holds if and only G = K n ¯ or G = K n .

Proof

Assume that d 1 , … , d n is the vertex degree sequence of G . We note that

(1) d i n ≤ d i n − d i ≤ d i .

Moreover, in the left-hand side, the equality holds if and only if d i = 0 and in the right-hand side, the equality holds if and only if d i = 0 or d i = n − 1 . Using (1) we find that

(2) ∑ i = 1 n d i n ≤ ∑ i = 1 n d i n − d i ≤ ∑ i = 1 n d i .

In the left-hand side, the equality holds if and only if d i = 0 , for i = 1 , … , n . Thus in the left-hand side, the equality holds if and only if G = K n ¯ . In the right-hand side, the equality holds if and only if d i = 0 or d i = n − 1 , for i = 1 , … , n . This shows that in the right-hand side, the equality holds if and only if G = K n ¯ or G = K n . Using (2), we obtain that

(3) 2 m n ≤ ∑ i = 1 n d i n − d i ≤ 2 m .

Moreover, in the left-hand side, the equality holds if and only if G = K n ¯ and in the right-hand side, the equality holds if and only G = K n ¯ or G = K n . This completes the proof.□

A real-valued function f is called convex if the line segment between any two distinct points on the graph of f lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. It is well known that a twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. We recall the well-known Jensen’s inequality related to the convex functions.

Theorem 2

(Jensen’s inequality) Let f be a one variable convex real-valued function. Then, for all real numbers x 1 , … , x n in the domain of f and all real numbers t 1 , … , t n , where 0 ≤ t i ≤ 1 and t 1 + ⋯ + t n = 1 , we have

f ( t 1 x 1 + t 2 x 2 + ⋯ + t n x n ) ≤ t 1 f ( x 1 ) + t 2 f ( x 2 ) + ⋯ + t n f ( x n )

and the equality holds if and only if x 1 = x 2 = ⋯ = x n or f is linear on a domain containing x 1 , … , x n .

Now, we obtain another lower bound for the temperature index of graphs that improves the lower bound of Theorem 1.

Theorem 3

Let G be a graph with n vertices and m edges. Then,

T ( G ) ≥ 2 m n n 2 − 2 m

and the equality holds if and only if G is regular.

Proof

Define the real function f as f ( x ) = x n − x . Thus, f ′ ( x ) = n ( n − x ) 2 and f ′ ′ ( x ) = 2 n ( n − x ) 3 (where f ′ ( x ) is the derivative of f with respect to x ). This shows that f ′ ′ > 0 on the interval ( − ∞ , n ) and so f is convex on this interval. Using Jensen’s inequality we obtain that

(4) f 1 n d 1 + ⋯ + 1 n d n ≤ 1 n f ( d 1 ) + ⋯ + 1 n f ( d n ) .

and the equality holds if and only if d 1 = ⋯ = d n , where d 1 , … , d n is the vertex degree sequence of G . Since d 1 + ⋯ + d n = 2 m and T ( G ) = f ( d 1 ) + ⋯ + f ( d n ) , we find that

(5) T ( G ) ≥ n f 2 m n ,

and by Theorem 2, the equality holds if and only if d 1 = ⋯ = d n . This implies that the equality holds if and only if G is regular. This completes the proof (we note that 2 m n n 2 − 2 m ≥ 2 m n and so this lower bound is better than the lower bound of Theorem 1).□

Remark 1

Let G be a graph with n vertices and m edges. Since for every real number x such that ∣ x ∣ < 1 ,

1 1 − x = 1 + x + x 2 + x 3 + ⋯ ,

we find that for 0 ≤ d < n ,

d n − d = d n 1 − d n = d n + d 2 n 2 + d 3 n 3 + ⋯ .

Assume that d 1 , … , d n is the vertex degree sequence of G . Since 0 ≤ d i ≤ n − 1 , we conclude that

(6) T ( G ) = ∑ i = 1 n d i n − d i = 1 n ∑ i = 1 n d i + 1 n 2 ∑ i = 1 n d i 2 + 1 n 3 ∑ i = 1 n d i 3 + ⋯ .

This shows that the series ∑ j = 1 ∞ 1 n j ∑ i = 1 n d i j is convergent to T ( G ) .

We continue this section by obtaining a lower bound and an upper bound for the temperature index of graphs in terms of the Zagreb index of graphs. For some other bounds for the first Zagreb index of graphs we refer to [8] and [14] and references therein.

Theorem 4

Let G be a graph with n vertices and m edges and δ = δ ( G ) . Then,

T ( G ) ≥ 2 m n + Z ( G ) n 2 − n δ ,

and the equality holds if and only if G is regular.

Proof

Assume that d 1 , … , d n is the vertex degree sequence of G (we recall that d 1 + ⋯ + d n = 2 m ). We note that for k ≥ 3 , ∑ i = 1 n d i k ≥ δ k − 2 ∑ i = 1 n d i 2 (that is ∑ i = 1 n d i k ≥ δ k − 2 Z ( G ) ) and the equality holds if and only if d i = δ or d i = 0 . Therefore, by (6) of Remark 1, we find that

(7) T ( G ) ≥ 2 m n + Z ( G ) 1 n 2 + δ n 3 + δ 2 n 4 + ⋯ ,

and the equality holds if and only if G is regular. On the other hand, since δ < n , 1 n 2 + δ n 3 + δ 2 n 4 + ⋯ is convergent to 1 n 2 − n δ . Thus,

T ( G ) ≥ 2 m n + Z ( G ) n 2 − n δ ,

and the equality holds if and only if G is regular.□

Theorem 5

Let G be a graph with n vertices and m edges and Δ = Δ ( G ) . Then,

T ( G ) ≤ 2 m n + Z ( G ) n 2 − n Δ ,

and the equality holds if and only if G is a disjoint union of a regular graph and some isolated vertices.

Proof

Assume that d 1 , … , d n is the vertex degree sequence of G . We note that for k ≥ 3 , ∑ i = 1 n d i k ≤ Δ k − 2 ∑ i = 1 n d i 2 (that is ∑ i = 1 n d i k ≤ Δ k − 2 Z ( G ) ) and the equality holds if and only if d i = Δ or d i = 0 . Thus, we obtain that

(8) T ( G ) ≤ 2 m n + Z ( G ) 1 n 2 + Δ n 3 + Δ 2 n 4 + ⋯ ,

and the equality holds if and only if G is a disjoint union of a regular graph with some isolated vertices. Since Δ < n , we conclude that

(9) T ( G ) ≤ 2 m n + Z ( G ) n 2 − n Δ ,

and the equality holds if and only if G is a disjoint union of a regular graph with some isolated vertices.□

We complete this section by finding a new lower bound for the Zagreb index of graphs in terms of the number of vertices, the number of edges, and the maximum degree of vertices of graphs.

Theorem 6

Let G be a graph with n vertices and m edges and Δ = Δ ( G ) . Then,

Z ( G ) ≥ 4 m 2 n 2 − 2 m ( n − Δ ) ,

and the equality holds if and only if G is regular.

Proof

The result follows by combining Theorems 3 and 5.□

3 Distribution of eigenvalues

In this section, we find some relations between the temperature index of graphs and their eigenvalues. First, we recall some well-known results.

Lemma 1

[1] Let G be a graph with vertex set { v 1 , … , v n } and adjacency matrix A. Then, the ( i , j ) ’s entry of A k is the number of walks of G with length k between v i and v j (from v i to v j ).

Theorem 7

[1, Rayleigh-Ritz ratio] Let B be a real symmetric matrix of size n. Assume that λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n are the eigenvalues of B. Then, for every non-zero vector X in R n we have

λ n ≤ X t B X X t X ≤ λ 1 .

In addition, in the left-hand side, the equality holds if and only if X is an eigenvector of B corresponding to λ n and in the right-hand side, the equality holds if and only if X is an eigenvector of B corresponding to λ 1 .

By j n , we mean the all-one vector of size n , in other words j n = [ 1 … 1 ︸ n ] t .

Lemma 2

Let G be a graph with adjacency matrix A. Assume that d 1 , … , d n is the vertex degree sequence of G and j = j n . Then,

j t A j = d 1 + ⋯ + d n = 2 m .
j t A 2 j = d 1 2 + ⋯ + d n 2 = Z ( G ) .

Proof

We note that A j = [ d 1 … d n ] t . Thus, j t A j = d 1 + ⋯ + d n = 2 m and so the first part is follows. For the second part note that j t A 2 j = j t A A j = ( A j ) t A j = d 1 2 + ⋯ + d n 2 = Z ( G ) and so the result follows.

Now, we obtain an interval that contains at least one eigenvalue of a graph.

Theorem 8

Let G be a graph with n ≥ 2 vertices and m edges and T = T ( G ) . Then, G has at least one eigenvalue in the interval − n − δ T − 2 m n , n − δ T − 2 m n . In fact, at least one eigenvalue of G, say λ , satisfies ∣ λ ∣ ≤ n − δ T − 2 m n .

Proof

By Theorem 4,

T ≥ 2 m n + Z ( G ) n 2 − n δ .

Thus, by the second part of Lemma 2,

(10) T ≥ 2 m n + j t A 2 j n 2 − n δ = 2 m n + 1 n − δ j t A 2 j n = 2 m n + 1 n − δ j t A 2 j j t j ,

where A is the adjacency matrix of G . Suppose that λ 1 , … , λ n are the eigenvalues of A such that ∣ λ 1 ∣ ≥ ⋯ ≥ ∣ λ n ∣ . By Rayleigh-Ritz ratio (Theorem 7), j t A 2 j j t j ≥ λ n 2 . Hence, by (10), we conclude that

T ≥ 2 m n + λ n 2 n − δ .

This implies that − n − δ T − 2 m n ≤ λ n ≤ n − δ T − 2 m n . The proof is complete.□

Now, we obtain a new lower bound for the spectral radius of graphs.

Theorem 9

Let G be a graph with n vertices and m edges. Let Δ = Δ ( G ) , λ = λ max ( G ) , and T = T ( G ) . Then,

λ ≥ n − Δ T − 2 m n .

Moreover, the equality holds if and only if G is regular.

Proof

Assume that d 1 , … , d n is the vertex degree sequence of G . By Theorem 5,

(11) T ( G ) ≤ 2 m n + Z ( G ) n 2 − n Δ ,

and the equality holds if and only if G is a regular with some isolated vertices. Thus, by the second part of Lemma 2,

(12) T ( G ) ≤ 2 m n + 1 n − Δ j t A 2 j n ,

where A is the adjacency matrix of G and the equality holds if and only if G is a regular with some isolated vertices. On the other hand, by Theorem 7, j t A 2 j n ≤ λ 2 . Hence by (12),

(13) T ( G ) ≤ 2 m n + λ 2 n − Δ ,

and the equality holds if and only if G is regular. This completes the proof.□

mr_oboudi@shirazu.ac.ir

Acknowledgments

The author is grateful to the referees for their helpful comments.

Funding information: Author states no funding involved.
Author contributions: The sole author is responsible for the entire work and has read and approved the finalmanuscript.
Conflict of interest: Author states no conflict of interest.
Data availability statement: Not applicable.

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Received: 2025-07-03

Revised: 2025-10-25

Accepted: 2025-11-03

Published Online: 2025-12-09

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temperature index; eigenvalues of graphs; spectral radius; Zagreb index

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