A note on a walk-based inequality for the index of a signed graph

A signed graph Ġ is a pair (G, σ), where G = (V , E) is an unsigned graph, called the underlying graph, and σ : E −→ {−1, +1} is the sign function. We denote the number of vertices of a signed graph by n. The edge set of a signed graph is composed of subsets of positive and negative edges. Throughout the paper we interpret an unsigned graph as a signed graph with all the edges being positive. The n×n adjacencymatrix AĠ of Ġ is obtained from the standard (0, 1)-adjacencymatrix ofG by reversing the sign of all 1s which correspond to negative edges. The largest eigenvalue of AĠ is called the index of Ġ and denoted by λ1. A detailed introduction to spectra of signed graphs can be found in [3]. Spectra of signed graph have received a great deal of attention in the recent years. In particular, some upper bounds for λ1 appeared in our previous works [1, 2]. In this note we generalize the result of [2] concerning an upper bound for λ1 in terms of certain standard invariants. Additional terminology and notation are given in Section 2. Our contribution and some consequences are given in Section 3.


Introduction
A signed graphĠ is a pair (G, σ), where G = (V , E) is an unsigned graph, called the underlying graph, and σ : E −→ {− , + } is the sign function. We denote the number of vertices of a signed graph by n. The edge set of a signed graph is composed of subsets of positive and negative edges. Throughout the paper we interpret an unsigned graph as a signed graph with all the edges being positive.
The n×n adjacency matrix AĠ ofĠ is obtained from the standard ( , )-adjacency matrix of G by reversing the sign of all 1s which correspond to negative edges. The largest eigenvalue of AĠ is called the index ofĠ and denoted by λ . A detailed introduction to spectra of signed graphs can be found in [3].
Spectra of signed graph have received a great deal of attention in the recent years. In particular, some upper bounds for λ appeared in our previous works [1,2]. In this note we generalize the result of [2] concerning an upper bound for λ in terms of certain standard invariants. Additional terminology and notation are given in Section 2. Our contribution and some consequences are given in Section 3.

Terminology and notation
If the vertices i and j are adjacent, then we write i ∼ j. In particular, the existence of a positive (resp. negative) edge between these vertices is designated by i We use d i to denote the degree of a vertex i ∈ V(Ġ); in particular, we write d + i and d − i for the positive and negative vertex degree (i.e., the number of positive and negative edges incident with i), respectively. For (not necessary distinct) vertices i and j, we use c ++ ij to denote the number of their common neighbours joined to both of them by a positive edge, c + ij to denote the the number of their common neighbours joined to i by a positive edge and to j by any edge. We also use the similar notation for all the remaining possibilities.
The de nition of a walk in a signed graph does not deviate from the same de nition in the case of graphs. So, a walk is a sequence of alternate vertices and edges such that consecutive vertices are incident with the corresponding edge. A walk in a signed graph is positive if the number of its negative edges (counted with their multiplicity if there are repeated edges) is not odd; otherwise, it is negative. In the same way we decide whether a cycle in a signed graph is positive or negative. We use w + r (i, j) and w + r (i) to denote the number of positive walks of length k starting at i and terminating at j and the number of positive walks of length k starting at i, respectively, and similarly for the numbers of negative ones.

Results
Our main result reads as follows.
where i is a vertex that corresponds to the largest absolute value of the coordinates of an eigenvector a orded by λ i , r (r ≥ ) is an integer, w + = w + r− (i, j), w − = w − r− (i, j) and n + i (resp. n − i ) is the number of vertices j such that w + ≠ (resp. w − ≠ ).
Proof. Let x = (x , x , . . . , xn) be an eigenvector associated with λ and let x i be the coordinate that is largest in absolute value. Without loss of generality, we may assume that x i = . Considering the ith and the jth equality of λ and By multiplying the equality (2) by w + = w + r− (i, j) and adding to (1), we get Observe that, for w + ≠ , the previous inequality reduces to Taking the summation over all j such that w + ≠ , we get Similarly, by multiplying the equality (2) by w − = w − r− (i, j) and subtracting it from (1), we get which, after taking the summation over all j such that w − ≠ , leads to Since by summing (3) and (4), we obtain which completes the proof.
The Laplacian matrix LĠ is de ned as LĠ = DĠ − AĠ, where DĠ is the diagonal matrix of vertex degrees.
Observe that the counterparts to (1) and (2) in the case of the Laplacian matrix LĠ are given by with the notations of Theorem 3.1.
For r = , we have n + , this is twice the sum of negative triangles passing through i. Thus Theorem 3.1 gives the upper bound obtained in [2]. For r = , we get as c +− ij + c −+ ij = w − , c ++ ij + c −− ij = w + . In particular case of graphs, the latter inequality reduces to λ d (i) + + λ ≤ (d (i) + )d i + w (i), where d (i) denotes the number of vertices at distance 2 from i (and then n + i = d (i) + ) and w (i) denotes the number of walks of length 3 starting at i.