Circulant graph and chordal ring
A circulant graph is a special case of a Cayley graph. Suppose that G(V, E) (V = {v
_{1}, v
_{2}, …,v
_{
m
}}, E = {e
_{1}, e
_{2}, …, e
_{
n
}}) is a graph with m vertices and n edges. The circulant graph can be defined as follows [2, 13]:

Definition 1: G(V, E) is a simple graph with m vertices and n edges. There are integers w
_{1}, w
_{2}, …, w
_{
j
} (w
_{1} < w
_{2} < … < w
_{
j
} < (m + 1)/2) that represent a jump sequence. Two vertices v_{
k
} and v_{
l
} of V are connected if and only if (k + w
_{
i
}) mod m = l, or (k – w
_{
i
}) mod m = l. This graph is defined as a circulant graph. A circulant graph with j jumps is usually denoted by CG(m; w
_{1}, w
_{2}, …, w
_{
j
}) or CG(m; W) with jump sequence W = {w
_{1}, w
_{2}, …, w
_{
j
}}, and W = j.

This definition does not guarantee that a circulant graph is connected. For example, CG(8; 1, 2) is a connected circulant graph, but CG(8; 2, 4) is not connected. Boesch and Tindell [5] found that a circulant graph is connected if and only if the greatest common divisor gcd(m, w
_{1}, w
_{2}, …,w
_{
j
}) is equal to 1. Such a graph is always known as a connected circulant graph. Furthermore, it has been proved that every connected circulant graph has a Hamiltonian cycle [17]. In particular, a complete graph can always be considered as a combination of all CG(m; w
_{
i
}) (1 ≤ i ≤ ⌊m/2⌋), which are considered as basic parts, as shown in Fig. 1.

In addition, the chordal ring network is introduced. There are two definitions for a chordal ring [13].

Definition 2: If a circulant graph CG(m, W) has w
_{1} equal to 1, it is known as a chordal ring. The edges with w
_{
i
} ≠ w
_{1} (2 ≤ i ≤ j) are denoted by chords of length w
_{
i
}.

Definition 3: G(V, E) is a simple graph with m vertices and n edges. Vertices v
_{1}, v
_{2}, …,v
_{
m
} in V are connected in sequence into a Hamiltonian cycle. There are integers w
_{1}, w
_{2}, …, w
_{
j
} (w
_{1} < w
_{2} < … < w
_{
j
} < (m + 1)/2). Two vertices v
_{
k
} and v
_{
l
} of V are connected if and only if k and l are odd numbers, and (k + w
_{
i
}) mod m = l. This graph is defined as a chordal ring. A circulant graph with j chords is usually denoted by CR(m, w
_{1}, w
_{2}, …, w
_{
j
}) or CR(m, W).

In this work, we consider the former definition of a chordal ring, i.e., Definition 2. From this definition, it can be deduced that a chordal ring is a special case of a connected circulant graph.
Average distance
The average distance \( \overline{D} \) is defined as the average length of the shortest paths between any two nodes in the network [2]. It is illustrated on the basis of bidirected graphs, which are essentially undirected graphs with edges represented by bidirected arrows instead of full lines [18]. \( \overline{D} \) can be expressed as
$$ \overline{D}=\frac{{\displaystyle {\sum}_{v_i\in V}{\displaystyle {\sum}_{v_j\in V,{v}_j\ne {v}_i}{D}_{ij}}}}{m\left(m1\right)} $$
(1)
D
_{
ij
} is the shortest path distance between vertices v
_{
i
} and v
_{
j
}. In general, the shortest path distance can be expressed as
$$ {D}_{ij}= \min {\displaystyle {\sum}_{e_{ij}\in E}{c}_{ij}{x}_{ij}} $$
$$ s.t.\kern1em {\displaystyle {\sum}_{j:{e}_{ij}\in E}{x}_{ij}}{\displaystyle {\sum}_{j:{e}_{ji}\in E}{x}_{ji}}=\kern0.5em \left\{\begin{array}{c}\hfill \kern0.5em 1,\hfill \\ {}\hfill 1,\hfill \\ {}\hfill \kern0.74em 0,\hfill \end{array}\right.\kern0.5em \begin{array}{c}\hfill i=s,\hfill \\ {}\hfill i=t,\hfill \\ {}\hfill \kern0.36em i\ne s,t.\hfill \end{array}\kern1em {x}_{ij}\ge 0 $$
(2)
In the definition of D
_{
ij
}, x
_{
ij
} denotes whether there is an edge e
_{
ij
} from v
_{
i
} to v
_{
j
} on the path from v
_{
s
} to v
_{
t
}, and c
_{
ij
} denotes the cost of edge e
_{
ij
}. They can be expressed as follows:
$$ {x}_{ij}=\left\{\begin{array}{c}\hfill 1\hfill \\ {}\hfill 0\hfill \end{array}\right.\kern2em \begin{array}{c}\hfill {e}_{ij}\; is\; on\; the\; path\; from\;{v}_s\;to\;{v}_t\kern0.84em \hfill \\ {}\hfill {e}_{ij}\; is\; not\; on\; the\; path\; from\;{v}_s\;to\;{v}_t\hfill \end{array} $$
(3)
$$ {c}_{ij}=\left\{\begin{array}{c}\hfill 1\kern0.48em \hfill \\ {}\hfill +\infty \hfill \end{array}\right.\kern1em \begin{array}{c}\hfill If\; there\; is\; an\; edge\; between\;{v}_i\; an d\;{v}_j\hfill \\ {}\hfill If\; there\; is\; no\; edge\; between\;{v}_i\; an d\;{v}_j\hfill \end{array} $$
(4)
According to the isomorphism of a circulant graph, \( \overline{D} \) can be simplified as [19–23]
$$ \overline{D}=\frac{{\displaystyle {\sum}_{v_j\in V,{v}_j\ne {v}_1}{D}_{1j}}}{m1} $$
(5)
A typical lower bound for the average distance is the Moore bound [2]. However, the Moore bound is attainable only for some special topologies [2, 6].
A circulant graph with a jump relatively prime with m is isomorphic to a chordal ring. Therefore, its average distance is equal to that of the corresponding chordal ring. In most cases, the optimal distance of a chordal ring is equal to that of a circulant with the same m. The smallest m that does not fit this rule is 450, according to the exhaustive research of Fiol [24, 25].
According to the definition of a circulant graph, there are C(⌊m/2⌋ , j) circulant graphs with different W for certain m and j. Correspondingly, the number of chordal rings for the same m and j is C(⌊m/2⌋ − 1, j − 1), which is j/⌊m/2⌋ the number of circulants on the same scale.
Connectivity ratio
Connectivity ratio is defined as the ratio of the number of reachable node pairs to the total number of node pairs in the network, and it can be calculated as\( C=\frac{{\displaystyle {\sum}_{i\in V}{\displaystyle {\sum}_{j\in V,j\ne i}{l}_{ij}}}}{m\left(m1\right)}, \) subject to
$$ {l}_{ij}=\left\{\begin{array}{c}\hfill 1\kern1em there\kern0.5em is\kern0.5em a\kern0.5em path\kern0.5em from\kern0.5em {v}_i\kern0.5em to\kern0.5em {v}_j\kern0.1em ;\kern0.6em \hfill \\ {}\hfill 0\kern1em there\kern0.5em is\kern0.5em no\kern0.5em path\kern0.5em from\kern0.5em {v}_i\kern0.5em to\kern0.5em {v}_j\kern0.1em .\hfill \end{array}\right. $$
(6)
The maximum network connectivity ratio for a network with q node failures can be expressed as
$$ {C}_{q, \max }=\frac{\left(mq\right)\left(mq1\right)}{m\left(m1\right)} $$
(7)
Assuming that the probability of q node failures is p
_{q}, we propose the probability weighted connectivity, calculated as \( \overline{C}={\displaystyle {\sum}_{q=1}^m{p}_q{C}_q} \)
\( \overline{C} \) is more suitable for measuring network survivability in the event of a disaster that may take a heavy toll.