Per-Spectral Characterizations of Bicyclic Networks

Spectral techniques are used for the study of several network properties: community detection, bipartition, clustering, design of highly synchronizable networks, and so forth. In this paper, we investigate which kinds of bicyclic networks are determined by their per-spectra. We find that the permanental spectra cannot determine sandglass graphs in general. When we restrict our consideration to connected graphs or quadrangle-free graphs, sandglass graphs are determined by their permanental spectra. Furthermore, we construct countless pairs of per-cospectra bicyclic networks.


Introduction
It was recognized in about last decade that graph spectra have several important applications in computer science.Graph spectra appear in internet technologies, pattern recognition, computer vision, data mining, multiprocessor systems, statistical databases, and many other areas.For example, spectral filtering is applied in the study of Internet structure [1].This method uses the eigenvectors of the adjacency and other graph matrices and some clusters in data sets represented by graphs.For more information about the applications of graph spectra in computer science see [2][3][4][5][6], among others.
Two graphs are per-cospectral if they share the same perspectrum.A graph  is said to be determined by its perspectrum (DPS for short) if there is no other nonisomorphic graph with the same per-spectrum.
Which graphs are determined by their adjacency spectra is an old problem in graph spectra theory.van Dam and Haemers [8,9] gave an excellent survey of answers to the question of which graphs are determined by the spectra of some graph polynomials.Merris et al. [10] first considered the problem which graph is DPS.And they showed that the five pairs adjacency cospectral graphs (see [11]) are DPS.Based on the result, they formulated that the per-spectrum seems a little better than the adjacency spectrum when it comes to distinguishing graphs which are not trees.In fact, characterizing what kinds of graphs are determined by the per-spectra is generally a very hard problem.Up to now, only a few types of graphs are proved to be DPS; see [12][13][14][15][16][17].
A bicyclic network is a simple connected graph in which the number of edges equals the number of vertices plus one [18].The sandglass graph is a bicyclic network, denoted by ( 3 ,  ℎ ,  3 ), obtained by appending a triangle to each pendant vertex of the path  ℎ .Lu et al. [19] proved that sandglass graphs are determined by their adjacency spectra.Motivated by the statement of Merris et al., a natural problem is whether sandglass graphs are determined by their perspectra.In this paper, we give a solution of this question.In what follows, we begin with some definitions and notions.Let ∪ be the union of two graphs  and  which have no common vertices.For any positive integer , let  denote the union of  disjoint copies of graph .The path and cycle on  vertices are denoted by   and   , respectively.Let   () denote the number of -cycles in .
Let   and   be two vertex-disjoint cycles.Suppose that where  ≥ 1 and  = 1 means identifying V 1 with V  , the resulting graph (see Figure 1), denoted by ∞(, , ), is called ∞-graph.Let  +1 ,  +1 , and  +1 be three vertex-disjoint paths, where , ,  ≥ 1 and at most one of them is 1.Identifying the three initial vertices and terminal vertices of them, respectively, the resulting graph (see Figure 1), denoted by (, , ), is called -graph.Then bicyclic networks can be partitioned into two classes: the class of graphs which contain ∞-graph as its induced subgraph and the class of graphs which contain -graph as its induced subgraph.
A subgraph  of  is a Sachs subgraph if each component of  is a single edge or a cycle.Merris et al. [10] gave a modified Sachs formula to compute the coefficients of the permanental polynomials of graphs.
Lemma 1 (see [10]).Let  be a graph with (, ) = ∑  =0   () − .Then where the sum is taken over all Sachs subgraphs  of  on  vertices, and () is the number of cycles in .
Lemma 2 (see [13]).Let  be a graph with  vertices and  edges, and let ( 1 ,  2 , . . .,   ) be the degree sequence of .Then Lemma 3 (see [17]).Let  be a graph with  edges, and let   () denote the degree sum of the three vertices on th triangle in .Then Lemma 4 (see [13]).The following can be deduced from the permanental polynomial of a graph :

Sandglass Graphs Are DPS
In this section, we will give the solutions of the problem which sandglass graphs are DPS?Checking graph   depicted in Figure 2, direct computation yields (  , ) =  13 + 14 11 − 4 10 + 74 9 − 40 8 + 186 7 − 136 6 + 230 5 − 180 4 + 130 3 − 76 2 + 25 − 4 = (( 3 ,  9 ,  3 ), ).This implies that the permanental spectra cannot determine sandglass graphs in general.Examining graph   again, we know that   is not connected and contains a quadrangle.It is natural to consider the problem which sandglass graphs are DPS when we restrict our consideration to connected graphs or quadrangle-free graphs, where the quadrangle-free graph is one which contains no quadrangles (i.e., cycles of length 4).We will answer these questions one by one in the following.
Theorem 8. Restricting consideration on quadrangle-free graphs, sandglass graphs are determined by their per-spectra.
Suppose that  is isomorphic to a graph which contains a sandglass graph ( 3 ,   ,  3 ) as its induced subgraph.Then  contains no quadrangles.By Lemma 7,  must be isomorphic to the sandglass graph ( 3 ,  ℎ ,  3 ).
In the following, we will prove that  is isomorphic to a graph containing  4 −  as its induced subgraph.
Suppose that  is even.By Lemma 6, we know that   (( 3 ,  ℎ ,  3 )) = 5.This implies, by Lemma 1, that  must have odd perfect matchings.Examining the structure of , we see that  has at most two perfect matchings.So,  only has uniquely one perfect matching.This implies that all triangles or 4-cycle in  are not a component of some Sachs subgraph of order .Thus, the perfect matching of  is a unique Sachs subgraph of order .By Lemma 1,   () = 1, which contradicts the fact that   (( 3 ,  ℎ ,  3 )) = 5.
Assume that  is odd.By Lemma 1 and examining the structure of , we know that the Sachs subgraphs of order  in  is only the union of a triangle and a perfect matching of  deleting all edges on the triangle.Then   () = 2.This contradicts   (( 3 ,  ℎ ,  3 )) = 5.
This completes the proof.
For any bicyclic network, it is difficult to discuss which is determined by its per-spectrum.We can construct countless pairs per-cospectral bicyclic networks.Let  be an arbitrary graph with a fixed vertex  and let   ⋅  denote the coalescence of  and  with respect to  and , which is the graph obtained from  ∪  by identifying  and .Similarly, we define   V ⋅ .Borowiecki [20] showed that if both  −  and   − V are per-cospectral, then both   ⋅  and   V ⋅  are also per-cospectral.As an example, let  =   be the bicyclic network depicted in Figure 3.As  −  and  − V are isomorphic, they are per-cospectral.By the above-mentioned result of Borowiecki [20], for any graph , both   ⋅  and  V ⋅  are per-cospectral.

Summary
Per-spectra is an important part of graph spectra.In this paper, we discuss properties of permanental spectra of bicyclic networks.We show that without some limitations bicyclic networks are not DPS.Particularly, we find a pair of per-cospectral graphs.Combining the result of Lu et al. [19], our results (Theorems 8 and 9) are beyond Merris et al. 's imagination.Finally, we pose the following conjecture.
Conjecture 10.Sandglass graphs with a perfect matching are DPS.