Starlike Block Graph Analysis

• Starlike block graphs are connected graphs where all clique blocks share a central cut vertex, forming a star-of-cliques structure.
• The squared distance matrix of these graphs admits explicit formulas for its characteristic polynomial, determinant, inverse, and inertia, aiding spectral analysis.
• Spectral radius extrema are determined by the distribution of noncentral vertices among blocks, with a unique positive eigenvalue driving key properties.

A starlike block graph, in the sense of "On the Squared Distance Matrix of a Starlike Block Graph" (Das et al., 13 Sep 2025), is a connected simple block graph with a central cut vertex contained in every block, so the graph is a star of cliques attached at one common articulation point. If the blocks are $K_{n_1+1},K_{n_2+1},\ldots,K_{n_b+1}$, all sharing the same central vertex, the graph is denoted $\mathcal{S}(n_1,n_2,\ldots,n_b)$ and has $n=1+\sum_{i=1}^b n_i$ vertices. The principal object studied for this class is the squared distance matrix $\Delta(G)=D(G)\circ D(G)$, where $D(G)$ is the distance matrix. For these graphs, the matrix admits exact formulas for its characteristic polynomial, determinant, inverse, inertia, and spectral-radius extremals (Das et al., 13 Sep 2025).

1. Definition, notation, and terminological scope

A block graph is a connected simple graph in which every block is a complete graph. A cut vertex is a vertex whose deletion disconnects the graph, and a block is a maximal connected subgraph with no cut vertex. In the class $\mathcal{S}(n_1,\ldots,n_b)$, every block contains the same central cut vertex, and each block contributes $n_i$ noncentral vertices together with the center. Thus the blocks intersect only in that central vertex (Das et al., 13 Sep 2025).

The notation

$\mathcal{S}(n_1,n_2,\ldots,n_b)$

refers to the starlike block graph whose blocks are

$K_{n_1+1},\,K_{n_2+1},\,\ldots,\,K_{n_b+1},$

all sharing the same central cut vertex, with

$n=1+\sum_{i=1}^b n_i.$

The term is not completely uniform across the literature. In "Fiedler vector analysis for particular cases of connected graphs" (Filho et al., 2020), a block-starlike graph is instead denoted

$\mathcal{S}(n_1,n_2,\ldots,n_b)$0

and consists of a central articulation vertex with several block-path branches attached, all blocks having the same clique size $\mathcal{S}(n_1,n_2,\ldots,n_b)$1. That usage is a block-graph analogue of a starlike tree, whereas $\mathcal{S}(n_1,n_2,\ldots,n_b)$2 is the more specialized star-of-cliques model in which each branch has only one block. This suggests that terminological care is required when comparing results across papers.

2. Squared distance matrix and explicit block form

For a connected graph $\mathcal{S}(n_1,n_2,\ldots,n_b)$3, the distance matrix is

$\mathcal{S}(n_1,n_2,\ldots,n_b)$4

and the squared distance matrix is its Hadamard square

$\mathcal{S}(n_1,n_2,\ldots,n_b)$5

Since $\mathcal{S}(n_1,n_2,\ldots,n_b)$6 is real symmetric, all its eigenvalues are real (Das et al., 13 Sep 2025).

For $\mathcal{S}(n_1,n_2,\ldots,n_b)$7, the distances take only a small set of values: a noncentral vertex is at distance $\mathcal{S}(n_1,n_2,\ldots,n_b)$8 from the center and from other vertices in the same clique block, while vertices in different blocks are at distance $\mathcal{S}(n_1,n_2,\ldots,n_b)$9 through the central cut vertex. Hence the squared distances are $n=1+\sum_{i=1}^b n_i$0 and $n=1+\sum_{i=1}^b n_i$1, and the matrix acquires a rigid block structure.

Let $n=1+\sum_{i=1}^b n_i$2, where $n=1+\sum_{i=1}^b n_i$3 is the singleton consisting of the central cut vertex and $n=1+\sum_{i=1}^b n_i$4 is the set of the $n=1+\sum_{i=1}^b n_i$5 noncentral vertices in block $n=1+\sum_{i=1}^b n_i$6. Then

$n=1+\sum_{i=1}^b n_i$7

The principal submatrix obtained by deleting the central vertex is denoted $n=1+\sum_{i=1}^b n_i$8 (Das et al., 13 Sep 2025).

This block form is the basis for the exact calculations that follow. In particular, the equitable partition $n=1+\sum_{i=1}^b n_i$9 leads to quotient matrices for both $\Delta(G)=D(G)\circ D(G)$0 and $\Delta(G)=D(G)\circ D(G)$1, and those quotient matrices determine the nontrivial spectral factors.

3. Characteristic polynomial, determinant, and cofactors

Using the equitable partition $\Delta(G)=D(G)\circ D(G)$2, the quotient-matrix characteristic polynomials are

$\Delta(G)=D(G)\circ D(G)$3

and

$\Delta(G)=D(G)\circ D(G)$4

A key structural fact is that $\Delta(G)=D(G)\circ D(G)$5 is an eigenvalue of both $\Delta(G)=D(G)\circ D(G)$6 and $\Delta(G)=D(G)\circ D(G)$7, with multiplicity exactly

$\Delta(G)=D(G)\circ D(G)$8

This arises from the $\Delta(G)=D(G)\circ D(G)$9-eigenvectors inside the blocks, of the form $D(G)$0, contributed by the submatrices $D(G)$1 (Das et al., 13 Sep 2025).

The full characteristic polynomials are therefore

$D(G)$2

and

$D(G)$3

Setting $D(G)$4 yields the determinant formulas

$D(G)$5

and

$D(G)$6

By the cofactor-determinant lemma of Bapat, the sums of all cofactors are

$D(G)$7

and

$D(G)$8

A notable corollary is

$D(G)$9

that is, the path of length $\mathcal{S}(n_1,\ldots,n_b)$0. This is the only case for which the later inverse representation fails.

4. Inertia and the inverse as a rank-one perturbation

The inertia analysis begins with $\mathcal{S}(n_1,\ldots,n_b)$1. Its quotient matrix $\mathcal{S}(n_1,\ldots,n_b)$2 is a rank-one perturbation of a diagonal matrix with negative diagonal entries, and the corresponding rank-one perturbation argument yields exactly one positive eigenvalue. Since $\mathcal{S}(n_1,\ldots,n_b)$3 is nonsingular,

$\mathcal{S}(n_1,\ldots,n_b)$4

Applying Haynsworth’s inertia additivity formula to

$\mathcal{S}(n_1,\ldots,n_b)$5

the Schur complement is

$\mathcal{S}(n_1,\ldots,n_b)$6

which is positive by the determinant and cofactor formulas. Hence

$\mathcal{S}(n_1,\ldots,n_b)$7

Accordingly, the squared distance matrix of every nontrivial starlike block graph has exactly one positive eigenvalue, no zero eigenvalues, and $\mathcal{S}(n_1,\ldots,n_b)$8 negative eigenvalues (Das et al., 13 Sep 2025).

The inverse is expressed through a Laplacian-like matrix. Define

$\mathcal{S}(n_1,\ldots,n_b)$9

together with the hatted products

$n_i$0

and similarly for $n_i$1, etc. Then

$n_i$2

and

$n_i$3

A symmetric matrix $n_i$4 is defined entrywise by

$n_i$5

and

$n_i$6

This matrix satisfies

$n_i$7

so it is Laplacian-like in the sense that all row and column sums are zero. The key identities are

$n_i$8

from which the inverse formula follows: $n_i$9

The matrix $\mathcal{S}(n_1,n_2,\ldots,n_b)$0 is positive semidefinite and has rank $\mathcal{S}(n_1,n_2,\ldots,n_b)$1: $\mathcal{S}(n_1,n_2,\ldots,n_b)$2 Its kernel contains $\mathcal{S}(n_1,n_2,\ldots,n_b)$3, the zero eigenvalue is simple, and all cofactors of $\mathcal{S}(n_1,n_2,\ldots,n_b)$4 are equal to

$\mathcal{S}(n_1,n_2,\ldots,n_b)$5

5. Extremal spectral radius for fixed $\mathcal{S}(n_1,n_2,\ldots,n_b)$6 and $\mathcal{S}(n_1,n_2,\ldots,n_b)$7

For fixed numbers of vertices $\mathcal{S}(n_1,n_2,\ldots,n_b)$8 and blocks $\mathcal{S}(n_1,n_2,\ldots,n_b)$9, the spectral radius $K_{n_1+1},\,K_{n_2+1},\,\ldots,\,K_{n_b+1},$0 is uniquely minimized and maximized by two explicit starlike block graphs. The unique minimizer up to isomorphism is

$K_{n_1+1},\,K_{n_2+1},\,\ldots,\,K_{n_b+1},$1

that is, one block is as large as possible and the remaining $K_{n_1+1},\,K_{n_2+1},\,\ldots,\,K_{n_b+1},$2 blocks are $K_{n_1+1},\,K_{n_2+1},\,\ldots,\,K_{n_b+1},$3’s. The unique maximizer up to isomorphism is

$K_{n_1+1},\,K_{n_2+1},\,\ldots,\,K_{n_b+1},$4

with the $K_{n_1+1},\,K_{n_2+1},\,\ldots,\,K_{n_b+1},$5 integers summing to $K_{n_1+1},\,K_{n_2+1},\,\ldots,\,K_{n_b+1},$6 and differing by at most $K_{n_1+1},\,K_{n_2+1},\,\ldots,\,K_{n_b+1},$7; equivalently, the block sizes are as equal as possible (Das et al., 13 Sep 2025).

The proof is organized around a transfer principle. If two block sizes satisfy $K_{n_1+1},\,K_{n_2+1},\,\ldots,\,K_{n_b+1},$8, then replacing

$K_{n_1+1},\,K_{n_2+1},\,\ldots,\,K_{n_b+1},$9

increases the spectral radius. In the language of the characteristic polynomial, the graph with more balanced block sizes has larger spectral radius at positive $n=1+\sum_{i=1}^b n_i.$0. Iterating this comparison produces the extremal ordering

$n=1+\sum_{i=1}^b n_i.$1

with equality only for the corresponding extremal graphs.

This balancing principle is specific to the squared distance matrix on the class $n=1+\sum_{i=1}^b n_i.$2. A plausible implication is that, within this class, large off-block entries equal to $n=1+\sum_{i=1}^b n_i.$3 reward a more even distribution of noncentral vertices across blocks, whereas concentration into one dominant block suppresses the Perron root.

6. Relation to adjacent spectral theories

The starlike-block-graph results sit near two distinct spectral traditions. One is the theory of starlike trees. In "No two starlikes have equal index" (Oliveira et al., 2017), a starlike is a tree with a unique vertex of degree $n=1+\sum_{i=1}^b n_i.$4, denoted $n=1+\sum_{i=1}^b n_i.$5, and every nonisomorphic starlike tree with $n=1+\sum_{i=1}^b n_i.$6 has a distinct adjacency index. More precisely, the ordering of starlike trees by largest adjacency eigenvalue coincides with the lexicographical ordering of the corresponding partitions of $n=1+\sum_{i=1}^b n_i.$7. The paper also identifies the smallest partition $n=1+\sum_{i=1}^b n_i.$8 and the balanced partition

$n=1+\sum_{i=1}^b n_i.$9

as the extremal endpoints for fixed $\mathcal{S}(n_1,n_2,\ldots,n_b)$00 and $\mathcal{S}(n_1,n_2,\ldots,n_b)$01. This provides a spectral template in which local transfers between branches govern global extremality.

The second tradition concerns Laplacian eigenvectors on block graphs. In (Filho et al., 2020), block-starlike graphs $\mathcal{S}(n_1,n_2,\ldots,n_b)$02 are introduced to study Fiedler vectors and algebraic connectivity. The paper proves that vertices inside a clique block that are true twins satisfy $\mathcal{S}(n_1,n_2,\ldots,n_b)$03 for every Fiedler vector, that equal branch lengths $\mathcal{S}(n_1,n_2,\ldots,n_b)$04 force Case B with the center as the central vertex, and that the inequality

$\mathcal{S}(n_1,n_2,\ldots,n_b)$05

ensures Case A. It also shows that, for a block-path graph $\mathcal{S}(n_1,n_2,\ldots,n_b)$06 with $\mathcal{S}(n_1,n_2,\ldots,n_b)$07 odd, coalescing the unique center with a clique $\mathcal{S}(n_1,n_2,\ldots,n_b)$08 produces a graph in the block-starlike family without changing the algebraic connectivity: $\mathcal{S}(n_1,n_2,\ldots,n_b)$09

Taken together, these papers indicate that central-branch combinatorics repeatedly control extremal behavior across different matrices: adjacency matrices for starlike trees, Laplacians for block-path and block-starlike graphs, and squared distance matrices for $\mathcal{S}(n_1,n_2,\ldots,n_b)$10. The exact objects and proofs differ, but the recurring motifs are the same: a distinguished center, a branch-parameter description, structural symmetry within blocks, and monotonicity under branch transfers.